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+{"text":"\\section{Introduction}\n\n\\subsection{The subject}\n\nThis small review is based on papers \\cite{BGMS-75},\n\\cite{GKST-Springer}, \\cite{TESLA-04}. It deals with high-energy\n$\\gamma\\gamma$ collisions, this is a new and promising area in\nhigh-energy physics related to fundamental problems of strong and\nelectro-weak interactions.\n\nOur knowledge about elementary particles and their interactions is\nmainly obtained from particle collisions. Accelerators with\ncolliding beams are called now {\\it colliders}. Most of fundamental\nresults in particle physics have been obtained from experiments at\nthe $pp,\\; p\\bar p, \\;e^+e^-$ and $ep$ colliders.\n\nPrincipal characteristics of colliders are:\n\\begin{enumerate}\n\\item{the {\\it energy} in the center-of-mass system (c.m.s)\n$E_{\\rm cm}$;}\n\\item{{\\it luminosity} of a collider $L$ which determines\ncollision rate $\\dot N$ of events with the cross section $\\sigma$ by\nrelation $\\dot N = L\\, \\sigma$;}\n\\item{{\\it types} of colliding particles.}\n\\end{enumerate}\n\nThe progress on high-energy colliders can be seen from Table\n\\ref{T:1}. Up to now and in the nearest future, the $pp$ and $ p\\bar\np$ colliders are the machines with the highest energy. That is why\nsuch epochal discoveries as $W$ and $Z$ bosons (responsible for weak\ninteraction) and $t$ quark had been performed at the S$p\\bar p$S and\nthe TEVATRON, respectively. It is not excluded that at future $pp$\ncollider LHC the Higgs boson $ H$ (thought to be responsible for the\norigin of the particle masses) will be discovered.\n\n\\begin{table}[th]\n\\centering\n \\caption{High energy colliders}\n\\begin{tabular}{lccl}\\hline\nCollider & Type & $ E_{\\rm cm} $, TeV & Start date \\\\  \\hline\n&&& \\\\[-2mm]\nS$p\\bar p$S & $p\\bar p$ & 0.6 & 1981 \\\\\nTEVATRON & $p\\bar p$ & 2 & 1987 \\\\\nLHC & $ p p$ & 14 & 2007 \\\\[2mm]\nHERA & $e p$ & 0.31 & 1992 \\\\[2mm]\nSLC & $e^+e^-$ & 0.1 & 1989 \\\\\nLEP-I & $e^+e^-$ & 0.1 & 1989 \\\\\nLEP-II &$ e^+e^- $ & 0.2 & 1999 \\\\[2mm]\nLinear collider & $e^+e^-$ & 0.5 & 2010? \\\\\nPhoton collider & $\\gamma \\gamma,\\; \\gamma e$ & 0.4 & 2010+?\n\\\\[2mm]\nMuon collider & $\\mu^+\\mu^-$ & 0.1$\\div$ 3 & ?? \\\\\n\\end{tabular}\n \\label{T:1}\n\\end{table}\n\nFor detail study of new phenomena, it is important not only the\nenergy but also types of colliding particles. The $e^+e^-$\ncolliders, being less energetic then $pp$ colliders, have some\nadvantages over proton colliders due to much lower background and\nsimpler initial state. Well known example --- the study of $Z$\nboson. It was discovered at $ p\\bar p$ collider S$p\\bar p$S where\nabout 100 events with $Z$ bosons were found among about $10^{11}$\nbackground events, while the detailed study of $Z$ boson had been\nperformed at the $e^+e^-$ colliders LEP and SLC which provided us\nwith more than $10^7$ $Z$-events with a very low background.\n\nAbout thirty years ago a new field of particle physics ---\nphoton-photon interactions --- has\nappeared~\\cite{Bal71}--\\cite{BKT}. Up to now $\\gamma \\gamma$\ninteractions were studied in collisions of virtual (or equivalent)\nphotons at $e^+e^-$ storage rings. It was obtained a lot of\ninteresting results. However, the number of the equivalent photons\nis by 2 order of magnitude less than the number of electrons.\n\nA new possibility in this field is connected with high-energy\n$e^{\\pm}e^-$ linear colliders which are now under development. The\nelectron bunches in these colliders are used only ones. This makes\npossible to ``convert'' electrons to real high-energy photons, using\nthe Compton back-scattering of laser light, and thus to obtain\n$\\gamma \\gamma$ and $\\gamma e$ colliders with real\nphotons~\\cite{GKST81a}--\\cite{GKPST84}. The luminosity and energy of\nsuch colliders will be comparable to those of the basic $e^{\\pm}e^-$\ncolliders.\n\nPhysical problems, which are now investigated in the $\\gamma \\gamma$\ncollisions, are mainly connected with strong interaction at large\n$\\sim \\hbar \/(m_\\pi c)$ and moderate small distances $\\sim \\hbar\n\/p_\\bot$, where $p_\\bot \\; \\stackrel{<}{\\sim} 10$ GeV\/c. In future\nit will be a continuation of the present day experiments plus\nphysics of gauge $(W^\\pm,\\; Z)$ bosons and Higgs $H$ bosons, i.e. it\nwill be problems of the electro-weak interactions, Standard Model\nand beyond, search of new particles and new interactions. In other\nwords, physics at high-energy $\\gamma \\gamma$ and $\\gamma e$\ncollisions will be very rich and no less interesting than at $pp$ or\n$e^+e^-$ collisions. Moreover, some phenomena can best be studied at\nphoton collisions.\n\nAt the end of this subsection it will be appropriate to cite some\nwords from the article ``Gamma-Ray Colliders and Muon Colliders'' of\nthe former President of the American Physical Society A.~Sessler in\n{\\it Physics Today}~\\cite{Sessler}:\n\n\\begin{quotation}\n\nIn high-energy physics, almost all of the present accelerators are\ncolliding-beam machines.\n\nIn recent decades these colliders have produced epochal discoveries:\nStanford SPEAR electron-positron collider unveiled the charmed-quark\nmeson and $\\tau$ lepton in 1970s. In the realm of high-energy\nproton-antiproton colliders, the Super Proton Synchrotron at CERN\ngave us the W$^{\\pm}$ and Z$^0$ vector bosons of electroweak\nunification in 1990s, and in 1999s the Tevatron at Fermilab finally\nunearthed the top quark, which is almost 200 times heavier than the\nproton.\n\n...What about other particles? Beam physicists are now actively\nstudying schemes for colliding photons with one another and schemes\nfor colliding a beam of short-lived $\\mu^+$ leptons with a beam of\ntheir $\\mu^-$ antiparticles.\n\nIf such schemes can be realized, they will provide extraordinary new\nopportunities for the investigation of high-energy phenomena.\n\nThese exotic collider ideas first put forward in Russia more that 20\nyears ago...\n\n\\end{quotation}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=8cm,angle=0]{f1_1.eps}\n\\caption\nFeynman diagrams for the elastic $\\gamma \\gamma$  scattering in QED}\n \\label{F1.1}\n\\end{figure}\n\n\\subsection{Interaction of photons in the Maxwell theory and QED }\n\nThe Maxwell's equations are linear in the strengths of the electric\nand magnetic fields. As a result, in the classical Maxwell theory of\nelectromagnetism, rays of light do not interact with each other. In\nquantum electrodynamics (QED) photons can interact via virtual\n$e^+e^-$ pairs. For example, an elastic $\\gamma \\gamma$ scattering\nis described by Feynman diagrams of Fig.~\\ref{F1.1}. The maximal\nvalue of the cross section is achieved at the c.m.s. photon energy\n$\\omega \\sim m_ec^2$ and is large enough:\n\\begin{equation}\n\\max \\;\\sigma_{\\gamma \\gamma \\to \\gamma \\gamma} \\sim \\alpha^4\n\\left({\\hbar\\over m_ec}\\right)^2= \\alpha^2 r^2_e\\sim 4\\cdot\n10^{-30}\\mbox{ cm}^2. \\label{1.3}\n\\end{equation}\nHowever, at low energies, $\\omega \\ll m_ec^2$, this cross section is\nvery small\n\\begin{equation}\n\\sigma_{\\gamma \\gamma \\to \\gamma \\gamma}=0.031 \\alpha^2 r^2_e\n\\left({\\omega\\over m_ec^2}\\right)^6\\,. \\label{1.1}\n\\end{equation}\nFor example, for visible light, $\\omega \\sim 1$ eV,\n\\begin{equation}\n\\sigma_{\\gamma \\gamma \\to \\gamma \\gamma} \\sim 10^{-65} \\; \\mbox{cm}\n^2\\,. \\label{1.2}\n\\end{equation}\nIt is too small to be measured even with the most powerful modern\nlasers, though there were such attempts. In recent\npaper~\\cite{Bernard} it was obtained an upper limit of the cross\nsection of $\\sigma(\\gamma \\gamma \\to \\gamma \\gamma)_{\\rm\nLim}=1.5\\times 10^{-48}$ cm$^2$  for the photon c.m.s. energy $0.8$\neV (see Fig.~\\ref{cro} from ~\\cite{Bernard}).\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=9cm,angle=0]{cro.eps}\n\\caption{Elastic photon cross section as a function of photon c.m.s.\nenergy}\n \\label{cro}\n\\end{figure}\n\nAt energies $\\omega > m c^2$, two photons can produce a pair of\ncharged particles. The cross section of the characteristic process\n$\\gamma \\gamma \\to \\mu^+ \\mu^-$ (Fig.\\ref{F1.2}a) is equal to\n\\begin{equation}\n\\sigma_{\\gamma \\gamma \\to \\mu^+ \\mu^-} = 4\\pi r_e^2\n\\frac{m_{e}^2c^4}{s} \\, \\ln{s\\over m^2_\\mu c^4} \\;\\;\\;\\; {\\rm at}\n\\;\\;\\;\\; s=(2\\omega)^2 \\gg 4m^2_\\mu c^4\\,.\n \\label{1.5}\n\\end{equation}\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=8cm,angle=0]{f1_2.eps}\n\\caption\nFeynman diagrams for a) $\\gamma \\gamma \\to \\mu^+\\mu^-$ and b)\n$e^+e^- \\to \\mu^+\\mu^-$} \\label{F1.2}\n\\end{figure}\nIt is larger than the ``standard'' cross section for the production\nof the same pair in the $e^+ e^-$ collisions (Fig.\\ref{F1.2}b via a\nvirtual photon only)\n\\begin{equation}\n\\sigma_{e^+e^- \\to \\mu^+ \\mu^-} = \\frac{4}{3}\\,\\pi r_e^2\\,\n\\frac{m_{e}^2c^4}{s}\\,.\n \\label{1.5a}\n\\end{equation}\n\n\\subsection{Collisions of equivalent photons at $e^+ e^-$\nstorage rings}\n\nUnfortunately, there are no sources of intense high-energy photon\nbeams (like lasers at low energies). However, there is indirect way\nto get such beams --- to use equivalent photons which accompanied\nfast charged particles. Namely this methods was used during last\nthree decades for investigation of two-photon physics at $e^+e^-$\nstorage rings. The essence of the equivalent photon approach can be\nexplained in the following way~\\cite{Fermi,WW} (see also~\\cite{BLP}\n\\S 99). The electromagnetic field of an ultra-relativistic electron\nis similar to the field of a light wave. Therefore, this field can\nbe described as a flux of the {\\it equivalent} photons with energy\ndistribution $dn_\\gamma \/ d\\omega$. The number of these photons per\none electron with the energy $E$ is\n\\begin{equation}\ndn_\\gamma\\sim {2\\alpha\\over \\pi}\\;\\ln{E\\over \\omega}\\,\n{d\\omega\\over\\omega} \\label{1.6}\n\\end{equation}\nor approximately\n\\begin{equation}\ndn_\\gamma\\sim 0.03\\;{d\\omega\\over\\omega}\\,.\n \\label{1.7}\n\\end{equation}\nAt the $e^+e^-$ colliders the equivalent photons also collide and\ncan produce some system of\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=11cm,angle=0]{f1_3.eps}\n\\caption\nProduction of system $X$ a) by two equivalent photons $\\gamma^*$\nwith 4-momenta (energies) $q_1\\;(\\omega_1)$ and $q_2\\; (\\omega_2)$\nemitted by an electron and a positron and b) in the annihilation\nprocess $e^+e^- \\to X$ } \\label{F1.3}\n\\end{figure}\nparticles $X$ (see Fig.~\\ref{F1.3}a, $\\gamma^*$ denotes the\nequivalent photon)\n\\begin{equation}\ne^+ e^-\\to e^+ e^-\\gamma^\\ast\\gamma^\\ast\\to e^+ e^- X\\,. \\label{1.8}\n\\end{equation}\nThus, this process is directly connected with the subprocess\n$\\gamma^* \\gamma^* \\to X$. Strictly speaking, the equivalent photons\nare not real photons, they are virtual ones. The 4-momentum squared\nof such a photon $q_i^2$ (which is equal to $m^2 c^2$ for usual\nparticle) is not equal zero, $q_i^2 \\neq 0$. But for large part of\nthe cross section $|q^2_i|$ is very small, therefore, the most of\nequivalent photons are almost real.\n\nThe cross section for two-photon production of  $e^+ e^-$ in\ncollisions of two fast particles with charges $Z_1e$ and $Z_2e$,\ni.e. for the $Z_1 Z_2 \\to Z_1 Z_2 e^+ e^-$ process, was calculated\nby Landau and Lifshitz \\cite{LL} in 1934 (see also \\cite{BLP} \\S\n100). In fact, it was the PhD of E.M.~Lifshitz.\n\nAt first sight, the cross sections of the two-photon processes at\n$e^+ e^-$ colliders (Fig.~\\ref{F1.3}a) are very small since they are\nthe 4-order processes: $ \\sigma_{\\rm two-phot} \\propto \\alpha^4\\,, $\nwhile for the annihilation processes of Fig.~\\ref{F1.3}b the cross\nsections $ \\sigma_{\\rm annih} \\propto \\alpha ^2. $ However, the\nannihilation cross sections decrease with increase of the energy\n(compare with (\\ref{1.5a}))\n\\begin{equation}\n\\sigma_{\\rm annih}\\sim{\\alpha^2}\\,{\\hbar^2 c^2\\over s},\\quad\ns=(2E)^2,\n \\label{1.9}\n\\end{equation}\nwhile the two-photon cross sections increase\n\\begin{equation}\n\\sigma_{\\rm two-phot}\\sim{\\alpha^4}\\,{\\hbar^2\\over m^2_{\\rm\nchar}c^2}\\, \\ln^n{s}\\,.\n \\label{1.10}\n\\end{equation}\nHere $n=3\\div 4$ depending on the process, and the characteristic\nmass $ m_{\\rm char}$ is constant (for example, $ m_{\\rm char}\\sim\nm_\\mu$ for $X= \\mu^+ \\mu^-$ and $ m_{\\rm char}\\sim m_\\pi$ for $X=\nhadrons$). As a result, already at $\\sqrt{s}> 2$ GeV\n\\begin{equation}\n\\sigma_{e^+ e^-\\to e^+ e^-\\mu^+\\mu^-} > \\sigma_{e^+\ne^-\\to\\mu^+\\mu^-}. \\label{1.11}\n\\end{equation}\nAnother  example, at the LEP-II electron-positron\n\\begin{figure}[htb]\n\\centering\n\\hspace*{5mm}\\includegraphics[width=11cm,angle=0]{f1_4.eps}\n\\caption{\nCross sections for some annihilation and two-photon processes in\n$e^+e^-$ collisions}\n \\label{F1.4}\n\\end{figure}\ncollider with the energy $\\sqrt{s}= 200$ GeV, the number of events\nfor two-photon production of hadrons with the c.m.s. energy\n$W_{\\gamma \\gamma} > 2$ GeV is by a three order of magnitude larger\nthan that in the annihilation channel (Fig.\\ref{F1.4}).\n\nAt $e^+e^-$ storage rings the first two-photon processes $e^+e^- \\to\ne^+e^- e^+e^-$ had been observed in 1970 (Novosibirsk~\\cite{Bal71}).\nThe importance of two-photon processes for the lepton and hadron\nproduction at $e^+e^-$ storage rings had been emphasized in the\npapers Arteage-Romero, Jaccarini, Kessler and Parisi~\\cite{Kes},\nBalakin, Budnev and Ginzburg~\\cite{BBG} and Brodsky, Kinoshita and\nTerazawa~\\cite{BKT}. In the papers~\\cite{BBG} it  was shown that\n$e^+e^-$ colliding beam experiments can give information about a new\nfundamental process $\\gamma^* \\gamma^* \\to hadrons$ and the\nnecessary formulae and estimations were obtained.\n\nAt that time there were a lot of theoretical investigations of\nvarious aspects of two-photon physics , but only a few experimental\nresults~\\cite{Bal71,Fras} have been obtained related mainly to the\nprocesses $\\gamma\\gamma \\to e^+ e^-$, $\\gamma\\gamma \\to\\mu^+ \\mu^-$.\nThis period of two-photon physics was summarized in review by\nBudnev, Ginzburg, Meledin and Serbo~\\cite{BGMS-75}.\n\nA few years later (approximately from 1977) it was shown in a number\nof theoretical papers that the two-photon processes are very\nconvenient for the test and detailed study of the Quantum\nChromodynamics (QCD) including investigation of\n\\begin{itemize}\n\\item a photon structure function (Witten \\cite{Witten}),\n\\item  a jet production in the $\\gamma \\gamma$ collisions\n(Llewelyn Smith \\cite{LS}; Brodsky, De Grand, Gunion and Weis\n\\cite{BGGW}; Baier, Kuraev and Fadin \\cite{BKF}),\n\\item  the $\\gamma \\gamma \\to c {\\bar c} c {\\bar c}$ process and\nthe problem of the perturbative Pomeron (Balitsky and Lipatov\n\\cite{BL}).\n\\end{itemize}\n\nA new wave of experimental activity in this field was initiated by\nthe experiment at SLAC~\\cite{ATel79} which demonstrated that\ntwo-photon processes can be successfully studied without detection\nof the scattered electrons and positrons. After that there was a\nflow of experimental data from almost all detectors at the $e^+e^-$\nstorage rings. It should be noted a special detector MD-1 in\nNovosibirsk with a transverse magnetic field in the interaction\nregion and system of registration of scattered at small angles\nelectrons which was developed for two-photon experiments (see review\n\\cite{MD1}). This period was reviewed by Kolanoski \\cite{Kol84}.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=12cm,angle=0]{savinov.eps}\n\\caption{Results for the process $\\gamma \\gamma \\to \\eta_c$}\n\\label{savinov}\n\\end{figure}\n\n\\section{Results obtained in virtual $\\gamma^* \\gamma^* $\ncollisions}\n\nIn experiments at $e^+ e^-$ storage rings a lot of interesting\nresults about $\\gamma^* \\gamma^* $ collisions have been obtained\n(see reviews \\cite{Kol84,Pope,MPW94} and Proceedings of Workshops on\nPhoton-Photon Collisions), among them:\n\\begin{itemize}\n\\item production of $C$-even resonances in $\\gamma ^* \\gamma ^*\n$ collisions, such as $\\pi ^0,\\; \\eta,$ $ \\eta ^{\\prime}, \\; f_2,\\;\na_2$, $ \\eta_c$, $\\chi_c$, ...  and measurement of their $\\gamma\n\\gamma$ width;\n\\item measurement of the total $\\gamma \\gamma \\to hadrons$\ncross section up to c.m.s. energy $W_{\\gamma \\gamma} $ about 150\nGeV;\n\\item measurement of the total $\\gamma ^* \\gamma ^* \\to hadrons$\ncross section with large values of $W^2_{\\gamma \\gamma} $ and photon\nvirtualities $-q^2_1 \\sim -q^2_2 \\sim 10$ (GeV\/c)$^2$;\n\\item a number of exclusive reactions: $\\gamma ^* \\gamma ^*\n\\to \\pi \\pi, \\; K \\bar K,\\; p \\bar p,\\; \\rho \\rho,\\; \\rho \\omega,$\netc.;\n\\item  investigation of the photon structure function in the\ncollision of almost real photon and highly virtual photon with\n$-q^2$ up to about 1000 (GeV\/c)$^2$;\n\\item jet production in $\\gamma \\gamma$ collisions.\n\\end{itemize}\nAs an example, let us present the beautiful result (Fig.\n\\ref{savinov}) for the study of two-photon process  $\\gamma \\gamma\n\\to \\eta_c$ at CLEO (Cornell).\n\nUnfortunately, the number of equivalent photons per one electron is\nrather small, and correspondingly the $\\gamma ^* \\gamma ^*$\nluminosity is about $3\\div 4$ orders of magnitude smaller than that\nin $e^+e^-$ collisions. Therefore, it is not surprising that the\nmost important results at $e^+e^-$ storage rings were obtained in\nthe $e^+e^-$ annihilation.\n\n\\section{Linear $e^{\\pm} e^-$ collider }\n\nNew opportunities for two-photon physics are connected with future\nlinear $e^{\\pm}e^-$ colliders. Projects of such accelerators are now\nunder development in several laboratories. A linear collider\nconsists of several main systems (see Fig. \\ref{schema-ses-ee}\nfrom~\\cite{Sessler}): electron injectors, pre-accelerators, a\npositron source, two damping rings, bunch compressors, main linacs,\ninteraction regions, a beam dump.\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=14cm,angle=0]{schema_ses_2.eps}\n\\end{center}\n \\caption{Schema of a linear $e^+e^-$ collider}\n  \\label{schema-ses-ee}\n \\end{figure}\n\nSince 1988 this field is developed in a very tight international\ncollaboration of physicists from many countries. In 1996-97 three\nprojects NLC (North America), JLC (Asia) and TESLA (Europe) have\npublished their Conceptual Design Reports~\\cite{Projects} of the\nlinear colliders in the energy range from a few hundred GeV to about\none TeV; in 2001 the TESLA Technical Design Report~\\cite{TESLA},\n\\cite{TESLA-04} has been published. One team at CERN is working now\non the conception of multi-TeV Compact Linear Collider (CLIC).\nCurrent parameters of these projects are presented in\nTable~\\ref{t2.1}. Parameters of the projects NLC and JLC are\npresented in one column since their teams developed a common set of\nthe collider parameters.\n\\begin{table}[!h]\n \\caption{Some parameters of the linear colliders  NLC\/JLC and TESLA}\n \\vspace{.2cm}\n\\renewcommand{\\arraystretch}{1} \\setlength{\\tabcolsep}{1.1mm}\n\\begin{center}\n\\begin{tabular}{lccc} \\hline &&NLC\/JLC&TESLA\n\\\\ \\hline \\hline\nC.m.s. energy $2E_0$& [TeV]& 0.5 & 0.5 \\\\\nLuminosity $L$& [$10^{34}$\/(cm$^{2}$s)]&  2.2 & 3 \\\\\nRepetition rate $f_r$ &[Hz]&  120 & 5 \\\\\nNo. bunch\/train $n_b$&  & 190& 2820\\\\\nNo. particles\/bunch $N_e$& [$10^{10}$] &  0.75 & 2 \\\\\nCollision rate $\\nu$& kHz &  22.8 & 14.1 \\\\\nBunch spacing $\\Delta t_b$& [ns] &  1.4 & 337  \\\\\nAccel. gradient $G$ & [MeV\/m]&  50 & $\\sim 25$\\\\\n{Linac length} $L_l$ & {[km]}& { 10} &{ 20} \\\\\nBeams power $2P_b$& [MW]& 14 & 22.5 \\\\\nIP beta-function $\\beta_x\/\\beta_y$& [mm] &   8\/0.1& 15\/0.4 \\\\\n{R.m.s. beam size at IP} $\\sigma_x\/\\sigma_y$ \\hspace{-5mm}\n& {[nm]}& {245\/2.7}& {555\/5} \\\\\nR.m.s. beam length $\\sigma_z$ & [$\\mu$]&  110&300 \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\label{t2.1}\n\\end{table}\n\nNow the project of International Linear Collider (ILC) is under\ndevelopment.\n\nA few special words have to be said about luminosity of the linear\ncolliders, which is determined as\n  \\begin{equation}\nL=\\nu\\;{N_{e^+}\\, N_{e^-}\\over S_{\\rm eff}}\\,,\n \\end{equation}\nwhere the effective transverse bunch aria $S_{\\rm eff}\\sim \\sigma_x\n\\sigma_y$. In the next table we present a comparison of the storage\nring LEP-II and linear colliders:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\nCollider & $ 2E_{e}$, &  $L$, $10^{33}$ & $\\nu$, & $N_{e^{\\pm}},$ &\n$\\sigma_x$, &$\\sigma_y$, \\\\ &  GeV &1\/(cm$^{2}$s) &\n kHz &  $10^{10}$ &\n$\\mu$ & nm\\\\  \\hline LEP-II & 200 & 0.05 & 45 & 30 & 200 & 8000\\\\\n\\hline NLC\/JLC & 500 & 22 & 22.8 & 0.75 & 0.245  & 2.7 \\\\\n\\hline TESLA   & 500 & 30  & 14.1   & 2    & 0,555 & 5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\nNote  transverse bunch sizes\n  $$\n{\\rm LEP-II}: \\;\\;\\;\\;\\;\\;\\sigma_x \\sigma_y \\;\\;\\sim \\;\\; 10^{-5}\n{\\rm cm}^2\\,,\n  $$\n $$\n{\\rm TESLA}: \\;\\; \\sigma_x \\sigma_y \\sim  3\\cdot 10^{-11} {\\rm\ncm}^2\\,.\n $$\n\n So, it is likely that a first linear collider  will have energy\nabout 500 GeV with some possible extension up to 1.5 TeV. Compared\nto the LEP, these colliders of the so called next generation are\ndesigned on 2.5--7 times higher energy and four orders of magnitude\nhigher luminosity!\n\n\n\\section{Photon collider on a base of a linear\n$e^{\\pm} e^-$ collider}\n\n\\subsection{Idea of high-energy $\\gamma\\gamma$ and\n$\\gamma e$ colliders with real photons}\n\n\nUnlike the situation in storage rings, {\\it in linear colliders each\n$e^{\\pm}$ bunch is used only once}. It makes possible to ``convert''\nelectrons to high-energy photons and to obtain the $\\gamma \\gamma$\nor $\\gamma e$ colliding beams with approximately the same energy and\nluminosity\\footnote{ This can not be done at usual $e^+e^-$ storage\nrings where a high luminosity is provided by large number of\ncollisions ($\\sim 10^9-10^{11} $) of the same $e^+$ and $e^-$\nbunches. Conversion of the electron and positron bunches into the\n$\\gamma $ bunches at the storage ring gives only a single collision\nof gamma bunches. The resulting luminosity will be very low because\nobtaining of new $e^+$ and $e^-$ bunches at storage rings takes long\ntime.} as in the basic $e^{\\pm}e^-$ collisions. Moreover, $\\gamma\n\\gamma$ luminosity may be even larger due to absence of some\ncollisions effects.\n\nThis idea was put forward by Novosibirsk group in 1981--1983\n(Ginzburg, Kotkin, Serbo and Telnov \\cite{GKST81a}--\\cite{GKST83})\nand was further developed in detail.\n\nAmong various methods of $e\\to \\gamma$ conversion (bremsstrahlung,\nondulator radiation, beamstrahlung and so on), the best one is the\nCompton scattering of laser light on high-energy electrons. In this\nmethod a laser photon is scattered backward taking from the\nhigh-energy electron a large fraction of its energy. The scattered\nphoton travels along the direction of the initial electron with an\nadditional angular spread $\\theta \\sim 1\/\\gamma_e$ where $\\gamma_e\n=E\/(m_e c^2)$ is the Lorentz factor of the electron. This method was\nknown long time ago~\\cite{ATM63} and has been realized in a number\nof experiments, e.g.~\\cite{FIAN64,SLAC69}. However, the conversion\ncoefficient of electrons to high-energy photons $k=N_\\gamma \/ N_e$\nwas very small in all these experiments. For example, in the SLAC\nexperiment \\cite{SLAC69} it was $\\sim 10^{-7}$.\n\nIn our papers \\cite{GKST81a,GKST83} it was shown that at future\nlinear $e^{\\pm}e^-$ colliders {\\it it will be possible to get}\n$k\\sim 1$ at a quite reasonable laser flash energy of a few Joules.\n\nTherefore, two principal facts, which make possible a photon\ncollider, are:\n\n\\begin{itemize}\n\n\\item linear colliders are single-pass accelerators, the\nelectron beams are used here only once;\n\n\\item obtaining of conversion coefficient $k\\sim 1$ is\ntechnically feasible.\n\n\\end{itemize}\n\nIt should be noted that {\\it positrons are not necessary for photon\ncolliders}, it is sufficient and much easier to use the\nelectron-electron colliding beams.\n\n\nThe problems of the $\\gamma \\gamma$ and $\\gamma e$ colliders were\ndiscussed on many conferences: Photon-Photon Collisions, Linear\nColliders, and dedicated $\\gamma \\gamma$ Workshops~\\cite{gg94,gg00}.\nVery rich physics, potentially higher than in $e^+e^-$ collisions\nluminosity, simplification of the collider (positrons are not\nrequired) are all attractive to physicists. Progress in development\nof linear $e^+e^-$ colliders and high power lasers (both\nconventional and free-electron lasers) makes it possible to consider\nphoton colliders as very perspective machines for investigation of\nelementary particles.\n\nThis option has been included in Conceptual~\\cite{Projects} and\nTechnical~\\cite{TESLA} Designs of linear colliders. All these\nprojects foresee the second interaction regions for the $\\gamma\n\\gamma$ and $\\gamma e$ collisions.\n\n\n\\subsection{Scheme of a photon collider}\n\n\nTo create a $\\gamma \\gamma$ or $\\gamma e$ collider with parameters\ncomparable to those in $e^+e^-$ colliders, the following\nrequirements should be fulfilled:\n\n\\begin{itemize}\n\\item the photon energy $\\omega \\approx E_0$ from $100$ GeV\nto several TeV;\n\\item the number of high-energy photons\n$N_\\gamma \\sim N_e \\sim 10^{10}$;\n\\item photon beams should be focused on the spot with transverse sizes\nclose to those which electron bunches would have at the interaction\npoint $\\sigma_x \\times \\sigma_y \\sim 10^{-5}$ cm $\\times 10^{-7}$\ncm.\n\\end{itemize}\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=11cm,angle=0]{f3_2.eps}\n\\caption{A principal scheme of a photon collider. High-energy\nelectrons scatter on laser photons (in the conversion region C) and\nproduce high-energy $\\gamma$ beam which collides with similar\n$\\gamma$ or $e$ beam at the interaction point IP.} \\label{F3.2}\n\\end{figure}\n\nThe best solution for this task is to use a linear $e^{\\pm}e^-$\ncollider as a basis and convert the $e^{\\pm}$ beams into $\\gamma$\nbeams by means of the backward Compton\nscattering~\\cite{GKST81a}---\\cite{GKPST84}.\n\nThe principal scheme is shown in Fig.~\\ref{F3.2}. An electron beam\nafter the final focus system is traveling towards the interaction\npoint IP. At the distance $b\\, {\\sim} 0.1 \\div 1$ cm from the\ninteraction point, the electrons collide with the focused laser beam\nin the conversion region C. The scattered high-energy photons follow\nalong the initial electron trajectories (with small additional\nangular spread $\\sim 1\/\\gamma_e$), hence,  {\\bf the high-energy\nphotons are also focused at the interaction point IP}. This very\nfeature is one of the most attractive point in the discussed scheme.\nThe produced $\\gamma $ beam collides downstream with the oncoming\nelectron or a similar $\\gamma $ beam.~\\footnote{To reduce\nbackground, the ``used\" electrons can be deflected from the\ninteraction point by an external magnetic field. In the scheme\nwithout magnetic deflection background is somewhat higher, but such\nscheme is simpler and allows to get higher luminosity.}\n\nMore details about the conversion region are shown in Fig.\n\\ref{conversion_ses_2} from~\\cite{Sessler}.\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[width=14cm,angle=0]{conversion_ses_2.eps}\n\\caption{Conversion region}\n \\label{conversion_ses_2}\n\\end{figure}\n\nIt is very important that modern laser technology allows to convert\nmost of electrons to high-energy photons. This means that the\n$\\gamma \\gamma$ luminosity will be close to the luminosity of the\nbasic $e^{\\pm}e^-$ beams.\n\n\n\\subsection{Compton scattering as a basic process for $e\\to\n\\gamma$ conversion}\n\nProperties of the linear and non-linear Compton scattering are\nconsidered in details in~\\cite{GKPST84}, \\cite{KPS98}, \\cite{IKS04}.\nIn the conversion region a laser photon with energy $\\omega_{0} \\sim\n1$ eV scatters on an electron with energy $E_0 \\sim 100$ GeV at a\nsmall collision angle $\\alpha _0$ (Fig.~\\ref{F6.1}) and produces a\nfinal photon with the energy $\\omega$ and the emission angle\n$\\theta$. Kinematics of the backward Compton scattering\n \\begin{equation}\n  e(p_0) +\\gamma_0 (k_0) \\to e(p) +\\gamma (k)\n \\label{6.1}\n  \\end{equation}\nis characterized by two dimensionless  variables $x$ and $y$:\n\\begin{equation}\nx\\;={2p_0 k_0\\over m^2c^2}\\approx {4E_0\\omega_{0}\\over m^{2}\nc^4}\\cos ^{2} {{\\alpha_{0}\\over 2}}\\,, \\;\\;\\; y\\; = {kk_0\\over\np_0k_0}\\approx {\\omega \\over E_0}\\,.\n \\label{6.2}\n\\end{equation}\n\\begin{figure}[htb]\n\\begin{center}\n\\vspace{-0mm}\n\\hspace*{-1cm}\\includegraphics[width=7cm,angle=0]{f6_1.eps}\n\\vspace{-0mm} \\caption{Kinematics of the backward Compton\nscattering} \\label{F6.1} \\vspace{-2mm}\n\\end{center}\n\\end{figure}\nThe maximum energy of the scattered photon $\\omega_{m}$ and the\nmaximum value of the parameter $y$ are:\n\\begin{equation}\n\\omega \\leq \\omega_{m}\\;= \\; {x\\over x+1}\\, E_0\\,, \\;\\;\\; y\\leq \\;\ny_m\\; =\\; {x\\over x+1} = {\\omega_{m}\\over E_0}\\;\\,. \\label{6.7}\n\\end{equation}\nThe energy of the scattered electron is $E= (1-y)E_0$ and its\nminimum value is $E_{\\min} = E_0\/(x+1)$.\n\nTypical example: in collision of the photon with $\\omega_0 = 1.17$\neV ($\\lambda =1.06\\;\\mu$m --- the region of the most powerful solid\nstate lasers) and the electron with $E_0= 250$ GeV, the parameter\n$x=4.5$ and the maximum photon energy\n  \\begin{equation}\n\\omega_m =0.82 E_0 = 205\\;\\;{\\rm GeV}\n \\end{equation}\nis close enough to the initial electron energy $E_0$.\n\nA photon emission angle is very small, $\\theta\\sim 1\/\\gamma_e =\n2\\cdot 10^{-6}$.\n\nThe total Compton cross section is\n $$\n\\sigma_{\\mathrm c}\\;=\\;\\sigma^{\\mathrm{up}}_{\\mathrm c}\\;+\n\\;2\\lambda_{e} P_{\\mathrm c} \\,\\tau_{\\mathrm c}\\, ,\n $$\n where $\\lambda_{e} $ is the mean helicity of the initial\nelectron, $P_c $ is that of the laser photon and\n \\begin{eqnarray}\n\\sigma^{\\mathrm{up}}_{{\\mathrm c}}&=&{2\\sigma_{0}\\over x}\n\\left[\\left(1-{4\\over x} - {8\\over x^{2}}\\right)\\ln (x+1) +{1\\over\n2} +{8\\over x} - {1\\over 2(x+1)^{2}}\\right] \\, ,\n \\label{6.8}\n \\\\\n \\tau_{\\mathrm c}&=& {2\\sigma_{0}\\over x} \\left[\\left(1+{2\\over\nx}\\right) \\ln (x+1)-{5\\over 2}+{1\\over x+1}-{1\\over\n2(x+1)^{2}}\\right]\\,,\n  \\nonumber\\\\\n  \\sigma_{0}&=& \\pi r_e^2=2.5\\cdot 10^{-25}\\;\\mbox {cm} ^{2}\\,.\n  \\nonumber\n \\end{eqnarray}\nNote, that polarizations of initial beams influence the total cross\nsection (as well as the spectrum) only if both their helicities are\nnonzero, i.e. at $\\lambda_{e} P_{\\mathrm c} \\neq 0$. The\nfunctions $\\sigma^{\\mathrm{up}}_{{\\mathrm c}}$, corresponding to the\ncross section of unpolarized beams, and $\\tau_{{\\mathrm c}}$,\ndetermining the spin asymmetry, are shown in Fig.~\\ref{F6.2}.\n\\begin{figure}[htb]\n\\begin{center}\n\\vspace{-9mm}\n\\includegraphics[width=10cm,angle=0]{f6_2.eps}\n\\vspace{-8mm} \\caption{Cross section $\\sigma^{\\mathrm{up}}_{{\\mathrm\nc}}$ for unpolarized photons and $\\tau_{{\\mathrm c}}$, related to\nspin asymmetry (see (\\ref{6.8}))}\n \\label{F6.2}\n\\end{center}\n\\end{figure}\nIn the region of interest $x= 1 \\div 5$ the  total  cross section is\nlarge enough\n \\begin{equation}\n\\sigma_{\\mathrm c}\\sim \\sigma_0=2.5\\cdot 10^{-25}\\;\\;{\\rm cm}^2\n \\label{6.12}\n \\end{equation}\nand only slightly depends  on  the  polarization of the initial\nparticles,  $|\\tau_{\\mathrm c}|\/\\sigma^{\\mathrm{up}}_{\\mathrm c}<\n0.1$.\n\nOn the contrary, the energy spectrum does essentially depend on the\nvalue of $\\lambda_{e} P_{\\mathrm c}$. The energy spectrum of\nscattered photons is defined by the differential Compton cross\nsection:\n \\begin{equation}\n{d\\sigma _{{\\mathrm c}}\\over dy} ={2\\sigma_{0}\\over x} \\left[{1\\over\n1-y} + 1-y - 4r(1-r) - 2\\lambda_{e} P_c\\;{y(2-y)\\over\n1-y}\\,(2r-1)\\right]\\,.\n \\label{6.13}\n \\end{equation}\nIt is useful to note that $r=y\/[x(1-y)] \\leq 1$ and $r\\to 1$ at\n$y\\to y_m$.\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-8mm}\n\\includegraphics[width=10cm,angle=0]{f6_3.eps}\n\\vspace{-8mm} \\caption{Energy spectrum of scattering photons at\n$x=4.8$} \\label{F6.3}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-5mm}\n\\hspace*{-0.5cm}\\includegraphics[width=7cm,angle=0]{f6_4a.eps}\\hspace*{-0.7cm}\n\\includegraphics[width=7cm,angle=0]{f6_4b.eps}\n\\vspace{-5mm}\n \\caption{Energy spectrum of scattering photons at\n$x=1$ (left figure) and $x=20$ (right figure)}\n \\label{F6.4}\n \\vspace{-2mm}\n\\end{center}\n\\end{figure}\nThe ``quality'' of the photon beam, i.e. the relative number of hard\nphotons, is better for the negative value of $\\lambda_{e}\nP_{\\mathrm c}$. For $2\\lambda_{e} P_c=-1$ the peak value of the\nspectrum at $\\omega =\\omega_{m}$ nearly doubles improving\nsignificantly the monochromaticity of the $\\gamma$ beam (cf. curves\n$a$ and $b$ in Fig.~\\ref{F6.3}).\n\nIn order to increase the maximum photon energy, one should use the\nlaser with larger frequency. This also increases a fraction of hard\nphotons (cf. Figs.~\\ref{F6.4}  and \\ref{F6.3}). Unfortunately, at\nlarge $x$ the high energy photons disappear from the beam producing\n$e^{+} e^{-}$ pairs in collisions with laser photons. The threshold\nof this reaction, $\\gamma\\gamma_L\\to e^+e^-$, corresponds to $x\n\\approx 4.8$. Therefore, it seems that the value $x\\approx 5$ is the\nmost preferable.\n\n\\subsubsection{Angular distribution}\n\nThe energy of a scattered photon depends on its emission angle\n$\\theta $ as follows:\n \\begin {equation}\n\\omega ={\\omega _m \\over 1+(\\theta \/\\theta_0)^2}; \\quad \\theta_{0}\n\\; = \\; {m c^2 \\over E_0}\\; \\sqrt{x+1}\\,.\n \\label{6.16}\n \\end{equation}\nNote, that photons with the maximum energy $\\omega_m$ scatter at\nzero angle. The angular distribution of scattered protons has a very\nsharp peak in the  direction of the incident electron momentum.\n\nAfter the Compton scattering, both electrons and photons travel\nessentially along the original electron beam direction. The photon\nand electron scattering angles ($\\theta$ and $\\theta_e$) are unique\nfunction of the photon energy:\n \\begin {equation}\n\\theta (y)=\\theta_0\\; \\sqrt{{y_m\\over y}-1}, \\;\\;\\; \\;\n\\theta_e={\\theta_0\\; \\sqrt{y(y_m-y)} \\over 1-y}\\; .\n \\label{6.21}\n \\end {equation}\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-4mm}\n\\includegraphics[width=8cm,angle=0]{f6_5.eps}\n\\vspace{-6mm} \\caption{ Photon and electron scattering angles vs the\nphoton energy $\\omega$  at $x=4.8$} \\label{F6.5} \\vspace{-6mm}\n\\end{center}\n\\end{figure}\nFor $x=4.8$ these  functions are plotted in Fig.~\\ref{F6.5}. It is\nremarkable that electrons are only slightly deflected from their\noriginal direction and scatter into a narrow cone:\n\\begin {equation}\n\\theta_e \\leq {x \\over 2\\gamma} = {2\\omega _0 \\over m c^2} .\n\\label{6.22}\n\\end {equation}\n\n\\subsubsection{Polarization of final photons}\n\nUsing the polarized initial electrons and laser photons, one can\nobtain the high-energy photons with various polarization. Let us\npresent two typical examples.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-8mm}\n\\includegraphics[width=9cm,angle=0]{f6_6.eps}\n\\vspace{-5mm} \\caption{Mean helicity of the scattered photons\n$\\lambda_\\gamma$ vs. $\\omega \/E_0$ for various laser photon\nhelicities $P_c$ and electron helicities $\\lambda _e$ at $x= 4.8$ }\n\\label{F6.6} \\vspace{-2mm}\n\\end{center}\n\\end{figure}\nThe mean helicity of the final photon $\\lambda_\\gamma$ in dependence\non the final photon energy $\\omega$ is shown in Figs.~\\ref{F6.6}.\nCurves $a,\\; b$ and $c$ in Fig.~\\ref{F6.6} correspond to spectra\n$a,\\; b$ and $c$ in Fig.~\\ref{F6.3}. In the case $2\\lambda_e\nP_c = -1$ (the case of good monochromaticity -- see curves $a$ in\nFig.~\\ref{F6.3}) almost all high-energy photons have a high degree\nof circular polarization. In the case $b$ the electrons are\nunpolarized, and the region, where high-energy photons have a high\ndegree of polarization, is much narrow.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\vspace{-8mm}\n\\includegraphics[width=9cm,angle=0]{f6_8.eps}\n\\vspace{-5mm}\n \\caption{Average linear polarization of the scattered\nphotons $l_\\gamma$  vs. $\\omega\/E_0$ at $x=1,\\; 2,\\; 3$ and $4.8$\n(the degree of linear polarization of the laser photon $P_l = 1$)}\n \\label{F6.8}\n\\end{center}\n\\end{figure}\nIf the  laser light has a linear polarization, then high-energy\nphotons are polarized in the same direction. The average degree of\nthe linear polarization of the final photon $l_\\gamma$ in dependence\non the final photon energy $\\omega$ is shown in Figs.~\\ref{F6.8}.\nThe linear polarization of the colliding photons is very important\nfor study of the nature of Higgs boson.\n\n\\section{Physics of $\\gamma\\gamma$ Interactions }\n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=14cm,angle=0]{all_cs.eps}\n \\caption{Cross sections for some interesting processes in\n $e^+e^-$, $\\gamma e$ and $\\gamma \\gamma$ collisions}\n \\label{all-cross}\n \\end{figure}\n\nPhysical potential of such $\\gamma \\gamma$ and $\\gamma e$ colliders\nwill be on the same level with future $e^+e^-$ and $pp$ colliders.\nMoreover, there is a number of problems in which photon colliders\nare beyond competition. The comparison of cross sections in the\n$e^+e^-$, $\\gamma \\gamma$ and $\\gamma e$ colliders are given in\nFig.~\\ref{all-cross}.\n\nPhoton collider (PC) makes it possible to investigate both problems\nof new physics and of ``classical\" hadron physics and QCD.\n\nSince photon couple directly to all fundamental charged particles\n--- leptons, quarks, $W$ bosons, super-symmetric particles, etc.\n--- a PC can provide a possibility to test every aspect of the\nStandard Model (SM) and beyond.\n\nBesides, photons can couple to neutral particles (gluons, $Z$\nbosons,  Higgs bosons, etc.) through charged particles box diagrams\n(Fig. \\ref{gg-to-higgs}).\n \\begin{figure}[!ht]\n \\centering\n \\includegraphics[width=7cm,angle=0]{gg-to-higgs.eps}\n  \\caption{Higgs boson production in $\\gamma \\gamma$ collision}\n   \\label{gg-to-higgs}\n  \\end{figure}\n\nOn the other hand, in a number of aspects photons are similar to\nhadrons, but with simpler initial state. Therefor, PC will be\nperfect in studying of QCD and other problems of hadron physics.\n\nLet us list the problems in which the photon colliders have a high\npotential or some advantages:\n\n{\\it Higgs hunting and investigation}. PC provides the opportunity\nto observe the Higgs boson at the smallest energy as a resonance in\nthe $\\gamma \\gamma$ system, moreover, PC is out of competition in\nthe testing of Higgs nature.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=15cm,angle=0]{ggtoSS.eps}\n \\caption{Cross sections for charged pair production in\n $e^+e^-$ and $\\gamma \\gamma$ collisions: here $S$ -- scalars, $F$ -- fermions,\n$\\sigma= (\\pi \\alpha^2\/M^2)\\, f(x)$}\n \\label{ggtoSS}\n\\end{figure}\n\n{\\it Beyond SM}. PC provides the excellent opportunities for finding\nof various particles beyond the SM: SUSY partners, charged Higgses,\nexcited leptons and quarks, leptoquarks,... In particular, $\\gamma\ne$ collider will be the best machines for discovery of selectron or\nexcited electron. Cross sections for the charged pair production in\n$\\gamma \\gamma$ collisions are larger than in $e^+e^-$ collisions\n--- see Fig. \\ref{ggtoSS}.\n\n{\\it Electroweak gauge boson physics}. The electroweak theory is the\nsubstantial part of the SM which pretend for precise description\nlike QED. PC will be $W$ factories with a rate about $10^7\\; W$\nbosons per year. In addition, the $\\gamma e \\to W \\nu$ process will\nproduce  single $W$s which is very attractive for $W$ decay's study.\nThus, PCs provide one of the best opportunity to test the precise\npredictions of the electroweak theory.\n\n{\\it QCD and hadron physics}. The photon colliders provide the\nunique possibility to investigate the problems of hadron physics and\nQCD in the new type of collisions and with the simplest structure of\ninitial state. The principal topics here are the following:\n\\begin{itemize}\n\\item{the $t \\bar t $ production in different partial waves;}\n\\item{the photon structure functions;}\n\\item{the semihard processes;}\n\\item{the jet production;}\n\\item{the total $\\gamma \\gamma \\to hadrons$ cross section.}\n\\end{itemize}\n\nTo clarify many of these points it will be very useful  to compare\nresults from $\\gamma \\gamma,\\; \\gamma e,\\; ep$ and $pp$ colliders.\n\nBesides the high-energy $\\gamma \\gamma $ and $\\gamma e$ collisions,\nPCs provide {\\it some additional options}:\n\n{\\it (i)} The region of conversion $e\\to \\gamma$ can be treated as\n$e\\gamma_L$ collider (here $\\gamma _L$ is the laser photon) with\nc.m.s. energy $\\sim 1 $ MeV but with enormous luminosity $\\sim\n10^{38}\\div10^{39}$ cm$^{-2}$s$^{-1}$.  It can be used, for example,\nfor search of weakly interacting light particles, like invisible\naxion.\n\n{\\it (ii)} In the conversion region one can test non-linear QED\nprocesses, like the $e^+e^-$ pair production in collision of\nhigh-energy photon with a few laser photons.\n\n{\\it (iii)} The used high-energy photon beams can be utilized for\nfixed-target experiments.\n\n\n\\section{Concluding remarks}\n\n\\subsection{Summary from TESLA TDR}\n\nThe TESLA Technical Design Report~\\cite{TESLA} contains the part\ndevoted to the high-energy photon collider~\\cite{TESLA-04}. It will\nbe useful to cite the  Physics Summary from this part:\n\n \\begin{quotation}\n\nTo summarize, the Photon Collider will allow us to study the physics\nof the electroweak symmetry breaking in both the weak-coupling and\nstrong-coupling scenario.\n\n Measurements of the two-photon Higgs width of the $h,\\,H$ and $A$\nHiggs states provide a strong physics motivation for developing the\ntechnology of the $\\gamma \\gamma$ collider option.\n\nPolarized photon beams, large cross sections and sufficiently large\nluminosities allow to significantly enhance the discovery limits of\nmany new particles in SUSY and other extensions of the Standard\nModel.\n\nMoreover, they will substantially improve the accuracy of the\nprecision measurements of anomalous W boson and top quark couplings,\nthereby complementing and improving the measurements at the $e^+e^-$\nmode of TESLA.\n\nPhoton colliders offer a unique possibility for probing the photon\nstructure and the QCD Pomeron.\n\n \\end{quotation}\n\n\\subsection{Prediction of Andrew Sessler~\\cite{Sessler} in 1998 }\n\n\\begin{quotation}\n\nAt present, Europe has the lead in electron colliders (LEP), hadron\ncolliders (LHC) and hadron-electron colliders(HERA). Stanford and\nJapan's High Energy Research Organization (KEK) are jointly working\non a TeV $e^+e^-$ collider disign, as in DESY. Japan and\/or Germany\nseem to be the most likely location for the next-generation $e^+e^-$\nmachine. Looking broadly, and also contemplating what US will do in\nhigh-energy physics, one may imagine a $\\mu^+\\mu^-$ collider in the\nUS, early in the next century.\n\n\\end{quotation}\n\n\\subsection{Conclusion of Karl von Weizs\\\"acker for young physicists}\n\nI would like to tell you a little real story. In 1991 the First\nInternational Conference on Physics devoted to Andrej Sakharov held\nin Moscow. A number of great man have participated in this\nConference including a dozen of Nobel prize winners. Among others\nwas Karl von Weizs\\\"acker.\n\nI personally was very interesting in application of the\nWeizs\\\"acker-Williams method to the two-photon processes at\ncolliding beams and I even have published a few articles on this\nsubject. So, I would like to see and to speak with the author of\nthis method. At the beginning of our conversation, Weizs\\\"acker told\nme a history of this invention.\n\nIn 1934 Weizs\\\"acker was in Copenhagen as an assistant of Prof.\nNiels Bohr. And just at that moment there was some international\nConference on Physics in the N.~Bohr Institute. Williams was the\nfirst who suggested the idea of the equivalent photon approximation\nfor QED processes. But the final result appeared only after a lot of\nheat discussions in which Weizs\\\"acker, Williams, Niels Bohr, Lev\nLandau, Edward Teller and some others participated.\n\nAfter the Conference all people, but Weizs\\\"acker, went back to\ntheir home institutes. So, it was quite natural that some day\nN.~Bohr invited  Weizs\\\"acker and said: ``And you, young man, you\nshould write a paper on the discussed subject''.\n\nAnd  Weizs\\\"acker did and brought the paper to Bohr.  A few days\nafter that Bohr invited him to discuss the work. ``At the beginning\nof this meeting, --- Weizs\\\"acker told me, --- I was young and\nexited, but Professor seemed to me was old and tired''.\n\nBohr said:``It is an excelent work, a very clean and perfect paper,\nbut... I have a small remark about the second page''. So they have\ndiscussed this small remark. After that they have discussed another\nremark, and another remark, and another...\n\n``After four or five hours of discussion. ---  Weizs\\\"acker\ncontinued, --- I was young and tired, but Professor was old and\nexcited''. At the end Bohr said:``Now I see that you wrote quite\ncontrary to what you think, you should rewrite your paper''.\n\n Weizs\\\"acker's conclusion was: ``I think that it is the best way for a\n young scientist to study physics: you should have a good problem\n and a possibility to discuss this problem during hours with a great man''.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:1}\n In a typical double-slit experiment interference fringes are formed\non a screen placed behind the slits which are then traversed by\nparticles of suitable wavelength.  By blocking one of the slits, the\nspatial interference pattern disappears.\n In a recent wave-vector space double slit experiment\n\\cite{pursehousedynamic} a photo-electron wave packet receding from\na single atom   is forced to pass through two energy levels within the atom, so that the levels play the role of the double-slit. As illustrated  in\nFig.\\,\\ref{fig0}, this is achieved by two low-intensity\ncontinuous-wave lasers which  excite the $5p$ and $6p$ states of a\nRubidium atom.  The infrared light field can concurrently ionize the\n$6p$ state and the blue laser the $5p$ state.   We image the interference in\nwave-vector space by scanning the photoelectron angular distribution.\nThe interference pattern disappears if only one state is excited,\ndemonstrating  that the phase relationship between the interfering\nwaves is imprinted by the atom. The two-lasers are not phase locked.\n\nThe $5p$ and $6p$ states thus represent  \"slits\", which can be closed by\ndetuning the respective laser fields, leading to non-resonant\nexcitation and damped occupation  of the specific intermediate state.\nThis \"damping\" is however different from thermal damping,\nas it does not swiftly destroy the coherence. The interference in\nthis case is affected because the two interfering amplitudes\nhave largely different strengths.\n\nIn the experiment the interference term and the associated\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig0}\n\\caption{(a) Schematic representation of the atomic \"double slit\nexperiment\" in Rubidium. The slits are represented by the bound and\ninitially unoccupied $5p$ and $6p$ intermediate states which are\nresonantly excited by the infrared and blue laser pulses. The two\nlasers subsequently ionize the intermediate levels, giving rise to two\nionization pathways $t_1$ and $t_2$. The photoelectron is transferred\ninto the continuum with a finite kinetic energy $E_f$ without knowing\nwhich pathway was taken. (b) Closing one slit by detuning one of the\ntwo lasers which then inhibits the occupation of the respective\nintermediate state (in this example the $6p$ state). We hence\nobtain a conventional two-color photoionization process via a single\nionization pathway. The energy level of the final state is now shifted down due\nto the detuning decreasing effectively $\\varepsilon_f=\\hbar\\omega_{\\rm BL}+\\hbar\\omega_{\\rm IR}+\\varepsilon_{5s}$.}\n\\label{fig0}\n\\end{figure}\ndifference in the phases of the amplitudes  were  recorded by\nessentially three measurements. First, both laser fields were set resonant\nto the $5p$ and $6p$ dipolar excitations leading to an ionization\namplitude $t_1+t_2$ [cf.\\,\\ref{fig0}(a)]. In the second and third\nmeasurements, one of the two laser fields was  detuned to\nclose one of the ionization pathways [cf.\\,\\ref{fig0}(b)] to extract\nthe individual amplitudes $t_1 (t_2)$. The ground state energy of the\n$5s$ Rb state is -4.177\\,eV while $E_{6p}=-1.589\\,$eV and\n$E_{5p}=-2.950\\,$eV. Thus, no other states in the bound spectrum of\nRb are accessible by single-photon processes.  An interesting point is\nthat in the experiment the two \\emph{cw} (continuous wave) lasers (cf. Fig.\\ref{fig0}) were\nnot phase locked and were weak. The calculations show that a\nrandom phase between these two lasers does not affect the\ninterference.  Thus the observed interference has to stem from the\natom, while the photoelectron wave is propagating out to the\ndetectors at infinity.\n\nThe question we are considering here is how the interference\nbehaves when the number of photons (the laser intensity) increases,\nand hence one would expect an increase in the probability that a\nblue and red photon are absorbed at the same time. The same process applies when\nwe consider  much shorter pulses. In this case one would expect the\nphase relation between the laser pulses to be important, and it then becomes possible\nto access the time scale on which the interference pattern  builds up and evolves.\n Unfortunately, due to experimental limitations we are currently not in a\nposition to investigate these ideas in the laboratory, and therefore the\ncurrent work is mostly theoretical.\n\nIn the final sections of this paper we discuss mechanisms for\ncontrolling and manipulating the interference phenomena between both\nionization pathways. Atomic units are used throughout the paper.\n\n\n\\section{Propagation on a space-time grid}\n\\label{sec:2}\n\nWithin the single-particle picture the Rubidium atom is well\ndescribed by the angular-dependent model potential introduced by\nMarinescu \\emph{et al.} \\cite{marinescu1994dispersion}:\n\\begin{equation}\nV_\\ell(r)=-\\frac{Z_\\ell(r)}{r} -\n\\frac{\\alpha_c}{2r^4}\\left[1-e^{-(r\/r_c)^6}\\right],\n\\end{equation}\nwhere $\\alpha_c$ is the static dipole polarizability of the\npositive-ion core and the effective radial charge $Z_\\ell(r)$ is\ngiven by\n\\begin{equation}\nZ_\\ell(r)=1+(z-1)e^{-a_1r}-r(a_3+a_4r)e^{-a_2r},\n\\end{equation}\nwith the nuclear charge $z$ and the cut-off parameter $r_c$ as well\nas the parameters $a_1-a_4$ fitted to experimental values. The\noptimized parameters are tabulated in\nRef.\\,\\cite{marinescu1994dispersion}.\n\nOn a very fine space grid ($\\Delta r=0.005$\\,a.u.) the radial wave\nfunctions of the atomic eigenstates\n${\\langle\\pmb{r}|\\phi_i\\rangle=R_{n_i,\\ell_i}(r)Y_{\\ell_i,m_i}(\\Omega_{\\pmb{r}})}$\nare found by (numerical) diagonalization of the matrix corresponding\nto the time-independent Hamiltonian\n$\\hat{H}_0=-(1\/2)\\partial_r^2+{\\ell(\\ell+1)\/(2r^2)} + V_\\ell(r)$. In\nthe presence of solenoidal and moderately intense electromagnetic\nfields the light-matter interaction Hamiltonian is given by $\\hat{H}_{\\rm\nint}(t)=-\\pmb{A}(\\pmb{r},t)\\cdot\\hat{\\pmb{p}}$ where\n$\\pmb{A}(\\pmb{r},t)$ is the vector potential and $\\hat{\\pmb{p}}$ is\nthe momentum operator.\n\nTo account for all multi-photon and multipole effects, a numerical\npropagation of the ground state wave function in the external vector\npotential is necessary, e.g. by the matrix iteration scheme\n\\cite{nurhuda1999numerical}. Exploiting the spherical symmetry of the\natomic system, the time-dependent wave function is decomposed in\nspherical harmonics, i.e. $\\Psi(\\pmb{r},t)=\\sum_{\\ell,m}^{\\ell_{\\rm\nmax}}b_{\\ell,m}(r,t)Y_{\\ell,m}(\\Omega_{\\pmb{r}})$. We therefore have to\npropagate $(\\ell_{\\rm max} + 1)^2$ channel functions\n$b_{\\ell,m}(r,t)$ which are coupled through the corresponding\n(dipole) matrix elements\n$\\langle\\ell'm'|\\pmb{A}\\cdot\\hat{\\pmb{p}}|m\\ell\\rangle$. Initially,\nthe ground state channel is fully occupied by the 5s Rubidium\norbital, i.e. ${\\Psi(\\pmb{r},t\\rightarrow-\\infty)=\n\\langle\\pmb{r}|\\phi_{5s}\\rangle}$ meaning\n${b_{0,0}(r,t-\\infty)=R_{n_i=5,\\ell_i=0}(r)}$. Introducing a time\n$T_{\\rm obs}$ where the external electromagnetic field perturbation\nis off, the wave function $\\Psi(\\pmb{r},t)$ is propagated to a\ntime $t>T_{\\rm obs}$ where the photoelectron wave packet is fully\nformed. The radial grid is extended to $10^4$\\,a.u. to avoid\nnonphysical reflections at the boundaries. Additionally, we\nimplemented absorbing boundary conditions by using an imaginary\npotential. The resulting simulation shows that the\nelectron density at the final grid point $r_N$ is then smaller than the\nnumerical error at the considered propagation times. \\\\\nTo obtain the scattering properties of the liberated electron, we\nproject  $\\Psi(\\pmb{r},T_{\\rm obs})$ onto a set of continuum wave\nfunctions which are given by the partial wave decomposition:\n\\begin{equation}\n\\langle\\pmb{r}|\\varphi_{\\pmb{k}}^{(-)}\\rangle=\\sum_{\\ell,m}i^\\ell\nR_{k\\ell}(r)e^{-i\\delta_\\ell(k)}\nY^*_{\\ell,m}(\\Omega_{\\pmb{k}})Y_{\\ell,m}(\\Omega_{\\pmb{r}}).\n\\label{eq:continuum}\n\\end{equation}\nHere, the kinetic energy is defined by $E_k = k^2\/2$,\n$\\delta_\\ell(k)$ are the scattering phases and $R_{k\\ell}(r)$ are\nradial wave functions satisfying the stationary radial Schr\u00f6dinger\nequation for positive energies in the same pseudopotential\n$V_\\ell(r)$ which is used for obtaining the bound spectrum. The\nscattering phases $\\delta_\\ell(k)={\\rm\narg}[\\Gamma(1+\\ell-i\/k)]+\\eta_\\ell(k)$ consist of the well-known\nCoulomb phases (first term) and phase $\\eta_\\ell(k)$ characterizing\nthe atomic-specific short-ranged deviation from the Coulomb\npotential. The radial wave functions are normalized to $\\langle\nR_{k\\ell}|R_{k'\\ell}\\rangle= \\delta(E_k-E_{k'})$. Finally, the\nprojection coefficients are given by\n\\begin{equation}\n\\begin{split}\na_{\\ell,m}(k) =&\\,e^{i(E_kT_{\\rm obs} + \\delta_\\ell(k) - \\ell\\pi\/2)}\n\\\\\n&\\times\\int_{r>r_a}{\\rm d}r\\,b_{\\ell,m}(r,T_{\\rm obs})R_{k\\ell}(r).\n\\end{split}\n\\end{equation}\nHere, we introduce the core radius $r_a$ and ensure that the\nintegration region is outside the residual ion. The photoionization probability (differential cross\nsection, abbreviated as DCS in the following) is defined as\n\\begin{equation}\n\\begin{split}\n{\\rm DCS} = \\frac{{\\rm d}\\sigma}{{\\rm d}\\Omega_{\\pmb{k}}}(E_k,\\Omega_{\\pmb{k}})\n\\propto &\\sum_{\\ell,\\ell'}\\sum_{m,m'} a^*_{\\ell',m'}(k)a_{\\ell,m}(k)\n\\\\ &\\times\nY^*_{\\ell',m'}(\\Omega_{\\pmb{k}})Y_{\\ell,m}(\\Omega_{\\pmb{k}}).\n\\end{split}\n\\end{equation}\nwhile the total cross section is\n$\\sigma(E_k)\\propto\\sum_{\\ell,m}\\sigma_{\\ell,m}(E_k)$ with\n$\\sigma_{\\ell,m}(E_k)=|a_{\\ell,m}(k)|^2$. Note that this treatment\ngives explicit insight into the population of the individual angular\nchannels and their contributions to the photoelectron wave packet.\n\nIn the following two-color ionization of Rubidium, the electric\nfields of both pulses are modeled according to\n\\begin{equation}\nE_p(t) =\n\\epsilon_p\\mathcal{E}_p\\Omega(t+\\Delta_p)\\cos(\\omega_p(t+\\Delta_p)+\\phi_p)\n\\end{equation}\nwith $p = {\\rm IR,BL}$ standing for the infrared and blue laser\npulses, respectively. The polarization vectors are $\\epsilon_p$, the\ntemporal envelope is given by $\\Omega(t)=\\cos^2(\\pi t\/T_d^p)$ for\n$t\\in[-T_d^p\/2,T_d^p\/2]$ with the pulse duration $T_d^p=2\\pi\nn_p\/\\omega_p$\ndetermined by the number of optical cycles $n_p$. Further we\nintroduce a temporal difference $\\Delta_p$ and a phase difference\n$\\phi_p$ to account for both laser fields originating from\ndifferent sources, and which are hence not phase-locked. Without loss of\ngenerality we set $\\phi_{\\rm IR}=0$ and $\\Delta_{\\rm IR}=0$. Both\nlaser fields are assumed to be linearly polarized in the $z$-direction\nso that the\nazimuthal angular quantum number $m$ is conserved. The number\nof angular channels then reduces to $\\ell_{\\rm max}+1$ and in the\nfollowing treatment we omit the subscript $m$ for brevity.\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig1}\n\\caption{Occupation numbers of the $5p$ and $6p$ intermediate states\nafter laser excitation, which depend on the time delay $\\Delta$ and\nthe phase difference  between both pulses. (a) shows the results of\nthe numerical propagation for six optical cycles. (b) corresponds to\nten optical cycles. Dots indicate the results for $\\phi=0$ radians and\ncrosses belong to $\\phi=2\/3\\pi$.}\n\\label{fig1}\n\\end{figure}\nTo balance the differences in the oscillator strengths between the\n$5s\\rightarrow5p$ and $5s\\rightarrow6p$ channels we used the field strengths\n$\\mathcal{E}_{\\rm BL}=0.05$ a.u. and $\\mathcal{E}_{\\rm IR}=0.007$\na.u. in the following simulations.\n\nThe aim here is to determine the influence of the phase\ndifferences on the resulting photoionization and occupation\nprobabilities. In Fig.\\,\\ref{fig1} we present the occupation numbers\nof the intermediate $5p$ and $6p$ states for two different laser\nconfigurations at a time $T_{\\rm obs}$ after laser excitation, meaning both\nlight pulses are completely extinguished and the photoelectron wave packet\npropagates freely in the Coulomb field of the residual Rubidium ion.\nPanel (a) corresponds to the case where both laser pulses have a\nlength of six optical cycles. We see that both occupation numbers\nshow a strong dependence on the temporal difference $\\Delta$ as well\nas on a random phase difference $\\phi$. The dots belong to $\\phi=0$\nwhile crosses indicate the results for a non-zero phase difference.\nHere, we show the occupation numbers for $\\phi=2\/3\\pi$, which show a\nlarge difference to the case of $\\phi=0$. In addition, we repeated\nthe simulation for other numbers of the random phase difference and\nobtained similarly pronounced discrepancies. The situation changes\ncompletely when increasing the number of optical cycles as shown in\npanel (b). For ten optical cycles the influence of both\nthe temporal and phase differences on the resulting occupation\nnumbers is drastically reduced, pointing to the transition into the\n\\emph{cw} limit.\n\nAlready from the \\emph{bounded} properties we suspect the rather fast\nconvergence of the photoionization process into a description\nwithin the frequency domain, which is characterized by infinitely long\nlaser pulses characterised by delta distribution-like bandwidths. We can\nunderline this observation by looking at the characteristics of the\nejected photoelectron wave packet. In Fig.\\,\\ref{fig2}(a) we present\nthe ionization probability of the angular channel $\\ell=2$\ncharacterized by the partial cross section\n$\\sigma_2=\\left|a_{\\ell=2}(k)\\right|^2$. As expected, the probability\ncurve sharpens under an increase in the pulse lengths of both laser\nfields. While for $n_p=6$ a manifold of energy states in the\ncontinuum is excited, from $n_p=10$ upwards we see clearly the\nunfolding of a Gaussian-like peak around the final energy\n$E_f=E_{5s}+\\hbar\\omega_{\\rm IR} + \\hbar\\omega_{\\rm BL}$. In the case\nof 35 optical cycles the FHWM of the probability peak is around\n0.07\\,eV.\n\nIt is also interesting to study the evolution of the quantum phase associated with\nthe photoelectron wave packet, which can be expressed as\n$\\varphi(k,\\vartheta_k)={\\rm\narg}\\left[\\sum_{\\ell}a_\\ell(k)Y_{\\ell,0}(\\Omega_{\\pmb{k}})\\right]$.\nFor 6 optical cycles, the angular channels with $\\ell=0$ and\n$\\ell=2$ already represent the dominant contributions to the photoelectron\nat the final energy around 0.4 eV, as expected for a two-color\nphotoionization process of an initial $s$-state. The quantum phase is\na result of interference between both partial waves and for the long\npulse limit it may be mathematically expressed by\n\\cite{watzel2014angular}\n\\begin{equation}\n\\varphi(k,\\vartheta_k)=\\arctan\\left[\\frac{\\sum_{\\ell=0,2}S_\\ell(k,\\vartheta_k)\\sin(\\phi_\\ell(k))}{\\sum_{\\ell=0,2}\nS_\\ell(k,\\vartheta_k)\\cos(\\phi_\\ell(k))}\\right]\n\\label{eq:phase}\n\\end{equation}\nwith\n$S_\\ell(k,\\vartheta_k)=\\left|a_\\ell(k)\\right|Y_{\\ell,0}(\\Omega_{\\pmb{k}})$\nand $\\phi_\\ell(k)={\\rm arg}[a_\\ell(k)]\\simeq\\delta_\\ell(k)-\\ell\\pi\/2$.\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig2}\n\\caption{Properties of the ejected photoelectron. (a) Ionization\nprobability of the angular channel $\\ell=2$ for different pulse\nlengths of the incident IR and blue laser fields. (b) Angular\nvariation of the averaged quantum phase\n$\\overline{\\varphi}(\\vartheta_k)$ in dependence on the number of\noptical cycles.}\n\\label{fig2}\n\\end{figure}\nThe quantum phase hence depends crucially on the scattering\nphases $\\delta_\\ell(k)$ and on the ratio between the transitions\nstrengths into $\\ell=2$ and $\\ell=0$ angular channels respectively.\nWe note that in principle $\\varphi(k,\\vartheta_k)$ is an experimentally accessible\nquantity, since it can be recovered by integrating the Wigner time\ndelay in photoionization defined as $\\tau_{\\rm\nW}(E_k,\\vartheta_k)=({\\rm d}\/{\\rm d}E_k)\\varphi(k,\\vartheta_k)$.\nThis can be extracted from delay measurements that are possible due\nto recent experimental advances within the attosecond timeframe\n\\cite{schultze2010delay, isinger2017photoionization}. The measured atomic time delay $\\tau_{\\rm a}$ consists of\nan \"intrinsic\" contribution (Wigner time delay $\\tau_{\\rm\nW}$) upon the absorption of an XUV photon which can be interpreted\nas the group delay of the outgoing photoelectron wave packet due to the collision process. As mentioned above, it contains information about the internal quantum phase. The second term $\\tau_{\\rm cc}$ arises from continuum-continuum transitions due to the interaction of the laser probe field\nwith the Coulomb potential and depends crucially on the experimental parameters. Hence, the difference $\\tau_{\\rm W}=\\tau_{\\rm a} - \\tau_{\\rm cc}$ provides access to the phase information $\\varphi(k,\\vartheta_k)$.\n\nIn Fig.\\ref{fig2}(b) it is shown how the\nphase develops by increasing the pulse lengths (number of optical\ncycles). Here, we show the phase averaged over the ionization probability (total cross section\n$\\sigma(E_k)$): $\\overline{\\varphi}(\\vartheta_k)=\\int{\\rm\nd}E_k\\,\\sigma(E_k)\\varphi(k,\\vartheta_k)\/\\int{\\rm\nd}E_k\\,\\sigma(E_k)$. Interestingly for very short pulses\n($n_p=6$) where the cross section is far from being centered\naround a final energy $E_f=E_{5s}+\\hbar\\omega_{\\rm IR} +\n\\hbar\\omega_{\\rm BL}$, the shape of the phase matches that extracted\nfrom simulations with longer pulses. As anticipated from earlier\nresults, from ten cycles upwards the results converge quickly into the\n\\emph{cw} limit. As an example, this is seen since the discrepancy between the quantum\nphase for 12 and 35 optical cycles is smaller than 5\\%.\n\n\\section{From short pulses to the \\emph{cw}-limit}\n\\label{sec:3}\n\n\\begin{figure*}[t!]\n\\includegraphics[width=0.95\\textwidth]{Fig3}\n\\caption{(a) Comparison between photoionization pathways via $5p$ and\n$6p$ intermediate states in the case of detuning $\\delta\\omega_{\\rm\nIR}=0.15$\\,eV of the infrared field. The curves are extracted from\nEq.\\,\\eqref{eq:d2_2}. (b) Time dependence of the first-order product\n$\\Re\\{[F_{\\omega_0}^{(1)}(t,-\\omega_{\\rm\nIR},0)]^*F_{\\omega_0}^{(1)}(t,-\\omega_{\\rm BL},\\Delta_{\\rm BL})\\}$\nfor different final energies. (c) Time dependent occupation numbers\n$c_{5p}(t)$ and $c_{6p}(t)$ extracted from perturbation theory and\na full numerical treatment. The grey curve indicates the temporal\nvariation of the IR laser field.}\n\\label{fig3}\n\\end{figure*}\n\n\nBy considering the two-color ionization process using perturbation\ntheory, we express the time-dependent wave function as\n$\\Psi(\\pmb{r},t)=\\sum_{\\nu\\ell} d_{\\nu\\ell}(t)e^{-{\\rm\ni}E_{\\nu\\ell}t}\\phi_{\\nu\\ell}(\\pmb{r})$. Note that the quantum number\n$\\nu$ includes both the bound and the continuum states. The first\norder amplitude is given by:\n\\begin{equation}\n\\begin{split}\nd^{(1)}_{0\\rightarrow\nf}(t)=&-\\frac{1}{i}\\sum_{\\lambda=\\pm1}\\left[\\langle f|D_{\\rm\nIR}|n\\rangle \\mathcal{E}_{\\rm\nIR}F_{\\omega_{0f}}^{(1)}(t,\\lambda\\omega_{\\rm IR},\\Delta=0) \\right. \\\\\n&\\left.+ e^{i\\lambda\\Delta\\omega_{\\rm BL}} \\langle f|D_{\\rm\nBL}|n\\rangle \\mathcal{E}_{\\rm\nBL}F_{\\omega_{0f}}^{(1)}(t,\\lambda\\omega_{\\rm\nBL},\\lambda\\Delta)\\right],\n\\end{split}\n\\label{eq:d1}\n\\end{equation}\nwhere $D_i=\\hat{\\epsilon}_i\\cdot\\hat{d}$ ($i={\\rm IR,BL}$) is the\ndipole operator, $\\omega_{0f}=E_f-E_0$ and\n\\begin{equation}\nF_{\\omega_{0f}}^{(1)}(t,\\omega,\\Delta)= \\int_{-\\infty}^{t}\n\\Omega(t'+\\Delta)e^{{\\rm i}(\\omega_{0f}+\\omega)t'} {\\rm d}t'.\n\\end{equation}\nWithout loss of generality we assume both laser fields are\ndescribed by the same temporal function $\\Omega(t)=\\cos^2[\\pi t\/T_p]$\nso that both pulses have the same pulse length\n$T_p=2n_p\\pi\/\\omega_{\\rm IR}$. In the following discussion the number of\noptical cycles $n_p$ hence refers to the infrared laser field. For the\nsquared-cosine shaped envelope $\\Omega(t)$ of the pulses, the\nfunction $F_{\\omega_{0}}^{(1)}(t,\\omega,\\Delta)$ can be obtained\nanalytically and converges against\n$F_{\\omega_{0}}^{(1)}(t,\\omega,\\Delta)\\rightarrow\\delta(\\omega_{0}-\\omega)$\nfor $T_p\\rightarrow\\infty$. The second order amplitude yields the\nfollowing expression:\n\\begin{equation}\n\\begin{split}\nd^{(2)}_{0\\rightarrow\nf}(t)=&-\\sum_{i,j}\\sum_{\\lambda,\\lambda'=\\pm1}\\sum_n\ne^{{\\rm i}(\\lambda\\Delta_j\\omega_j+\\lambda'\\Delta_i\\omega_i)}\\\\\n&\\times\\mathcal{E}_j\\mathcal{E}_i \\frac{\\langle f|D_j|n\\rangle\n\\langle n|D_i|0\\rangle}{\\omega_{0n}-\\omega_i} \\\\\n&\\times\nF^{(2)}_{\\omega_{nf},\\omega_{0n}}(t,\\lambda\\omega_j,\\lambda'\\omega_i,\\lambda\\Delta_j,\\lambda'\\Delta_i),\n\\end{split}\n\\label{eq:d2_2}\n\\end{equation}\nwhere again $i,j={\\rm IR,BL}$. The second-order temporal function is\ndefined as\n\\begin{equation}\n\\begin{split}\nF&^{(2)}_{\\omega_{nf},\\omega_{0n}}(t,\\omega_j,\\omega_i,\\Delta_j,\\Delta_i)=(\\omega_{0n}-\\omega_i)\\\\\n&\\times\\int_{-\\infty}^{t}{\\rm d}t'\\,\\Omega(t'+\\Delta_j)e^{{\\rm\ni}(\\omega_{nf}+\\omega_j)t'}F^{(1)}_{\\omega_{0n}}(t',\\omega_i,\\Delta_i).\n\\end{split}\n\\end{equation}\nA closed expression for the second-order temporal function $F^{(2)}$\ncannot be obtained analytically. However, a solution for $t>T_p\/2$\n(time of switch off) can be found and investigated for\n$T_p\\rightarrow\\infty$ (the continuous wave limit). It follows that in\nthe many optical cycle limit we find energy conservation, i.e.\n\\begin{equation}\n\\lim_{n_p\\rightarrow\\infty}\nF^{(2)}_{\\omega_{nf},\\omega_{0n}}(t,-\\omega_j,-\\omega_i,\\Delta_j,\\Delta_i)\n= \\frac{3\\pi}{4}\\delta(\\omega_{0f} - \\omega_{\\rm j} - \\omega_{\\rm i})\n\\label{eq:F2_asymptotic}\n\\end{equation}\nFurther, in this limit the individual temporal shifts $\\Delta_j$ and\n$\\Delta_i$ have no influence on the (second-order) transition\namplitude. Thus, in the long (\\emph{cw}) pulse limit and for energies around\n$E_f=E_0+\\hbar\\omega_{0f}$, this behavior allows us to rewrite the\nresulting second-order transition amplitude:\n\\begin{equation}\n\\begin{split}\nd^{(2)}_{0\\rightarrow\nf}\\underset{n_p\\rightarrow\\infty}{\\rightarrow}&\\frac{3\\pi}{4}\\mathcal{E}_{\\rm\nIR}\\mathcal{E}_{\\rm BL}e^{{\\rm i}\\Delta\\omega_{\\rm\nBL}}\\sum_n\\left[\\frac{\\langle f|D_{\\rm IR}|n\\rangle\\langle n|D_{\\rm\nBL}|0\\rangle}{\\omega_{n0}-\\omega_{\\rm BL}}\\right. \\\\\n&\\left.+ \\frac{\\langle f|D_{\\rm BL}|n\\rangle\\langle n|D_{\\rm\nIR}|0\\rangle}{\\omega_{n0}-\\omega_{\\rm IR}} \\right],\n\\end{split}\n\\label{eq:2PhotonMatrix}\n\\end{equation}\nwhich is similar to the traditional form of the two-photon matrix\nelement \\cite{toma2002calculation}. From here we see directly that a\nrandom phase $\\phi=\\Delta\\omega_{\\rm BL}$ does not play any role\n(especially when analyzing the cross sections\n$\\sim|d^{(2)}_{0\\rightarrow f}|^2$.\n\nLet us now come back to the two-color ionization process in the Rubidium\natom. It is crucial for the first and second-order amplitudes to\nprecisely evaluate the dipole matrix elements \\cite{amusia2013atomic}\n\\begin{equation}\n\\langle\nf|D_i|n\\rangle=(-1)^{m_f+\\ell_>}\\begin{pmatrix}\\ell_f&1&\\ell_n \\\\\n-m_f&0&m_n \\end{pmatrix}\\sqrt{\\ell_>}\\langle f||\\hat{d}||n\\rangle.\n\\end{equation}\nwhere $\\ell_>={\\rm max}(\\ell_f,\\ell_n)$ and the reduced radial matrix\nelement $d_{\\ell_f\\ell_n}=\\langle\nf||\\hat{d}||n\\rangle=\\int_0^\\infty{\\rm d}r\\,R_f(r)D_iR_n(r)$. \nOne way to account, at least partially, for the the interaction between the valence  and the core electrons \\cite{hameed1968core}\n is to modify the operator  $\\hat{Q}_L$ as \n\\begin{equation}\n\\hat{Q}_L\\rightarrow\\hat{Q}_L\\left[1-\\frac{a_c^{(L)}}{r^{2L+1}}\\left(1-e^{-(r\/r_c')^{2L+1}}\\right)\\right],\n\\label{eq:core_Q}\n\\end{equation}\nwhere $a_c^{(L)}$ is the  $2^L$ tensor core polarizability and $r_c'$ is an empirical cut-off radius (for Rb $r_c'=4.339773$ a.u.\n\\cite{marinescu1994dispersion}). This physical picture behind the corrections is roughly that  the valence electron with a dipole moment $\\pmb{d}$ induces  (by virtue of its field)  a (core) dipole moment $-\\alpha_c\\pmb{d}\/r^3$ (and higher multipoles).  Then, the complete dipole moment becomes $\\pmb{d}(1-\\alpha_c\/r^3)$. Note that in our case the Dipole operator $\\hat{d}=\\hat{Q}_1$. Using the modified dipole operator delivers very accurate matrix elements near the ionization threshold in comparison with experiments and more sophisticated theoretical models\n\\cite{petrov2000near}.\n\nWe learn from Eq.\\,\\eqref{eq:2PhotonMatrix} that changing the\nelectric field amplitudes $\\mathcal{E}_{\\rm IR}$ and\n$\\mathcal{E}_{\\rm BL}$ will not balance any differences between the\nmatrix element products $\\langle f|D_{\\rm IR}|n\\rangle\\langle\nn|D_{\\rm BL}|0\\rangle$ and $\\langle f|D_{\\rm BL}|n\\rangle\\langle\nn|D_{\\rm IR}|0\\rangle$. Thus, to reach equipollent ionization\npathways $E_{5s}\\rightarrow E_{5p}\\rightarrow E_f$ and\n$E_{5s}\\rightarrow E_{6p}\\rightarrow E_f$ we have to detune the laser\nfrequency corresponding to the stronger bound-bound transition. Given\nthe reduced matrix elements $\\langle 5p||\\hat{d}||5s\\rangle=-5.158$\nand $\\langle 6p||\\hat{d}||5s\\rangle=0.468$, we have to detune the\ninfrared field as shown in Fig\\,\\ref{fig3}(a). The curves are\nobtained from a numerical integration of Eq.\\,\\eqref{eq:d2_2}. For\n$n=35$ optical cycles, a detuning of\n$\\delta\\omega_{\\rm IR}=+0.15$\\,eV is required to allow the second-order\ntransition amplitudes within each pathway to have the same magnitude.\nNote that the value of $\\Delta\\omega_{\\rm IR}$ decreases by\nincreasing the pulse lengths. In this vein we performed the\nsimulation for 75 optical cycles and found that a detuning of only\n0.07 eV is needed to reach equipollent ionization pathways (not shown\nfor brevity).\n\nLet us consider the time-dependent first order probability\n$P_{0\\rightarrow f}^{(1)}(t)=\\left|d^{(1)}_{0f}(t)\\right|^2$ which\nreads explicitly\n\\begin{equation}\n\\begin{split}\nP_{0\\rightarrow f}^{(0)}(t)=&\\left|\\langle f|D_{\\rm IR}|n\\rangle\n\\mathcal{E}_{\\rm IR} F_{\\omega_{0f}}^{(1)}(t,-\\omega_{\\rm\nIR},0)\\right|^2 \\\\\n&+ \\left|\\langle f|D_{\\rm BL}|n\\rangle \\mathcal{E}_{\\rm\nBL}F^{(1)}_{\\omega_{0f}}(t,-\\omega_{\\rm BL},\\Delta_{\\rm BL})\\right|^2\n\\\\\n&+2\\mathcal{E}_{\\rm IR}\\mathcal{E}_{\\rm BL} \\langle f|D_{\\rm\nIR}|n\\rangle\\langle f|D_{\\rm BL}|n\\rangle \\\\\n&\\times\\Re\\left\\{F_{\\omega_{0f}}^{(1)}(t,-\\omega_{\\rm\nIR})\\left[F_{\\omega_{0f}}^{(1)}(t,-\\omega_{\\rm BL},\\Delta_{\\rm\nBL})\\right]^*\\right\\}.\n\\end{split}\n\\end{equation}\nHere, we have to emphasize that all terms containing $+\\omega_i$ are\nnegligibly small which we refer to as the rotating wave approximation.\nThe last term in the third line might look like a two-photon process\nbut a closer inspection of the function\n$F_{\\omega_0}^{(1)}(t,-\\omega,\\Delta)$ reveals that it sharply peaks\naround $\\omega_0-\\omega=0$ even for times close to $T_p\/2$. Since\n$\\omega_{f0}$ is fixed, the product between both $F^{(1)}$ functions\nis zero, so that\n\\begin{equation}\n\\begin{split}\n\\lim_{T_p\\rightarrow\\infty} F_{\\omega_{f0}}^{(1)}&(T_p\/2,-\\omega_{\\rm\nIR},0)\\left[F_{\\omega_{f0}}^{(1)}(T_p\/2,-\\omega_{\\rm BL},\\Delta_{\\rm\nBL})\\right]^* \\\\\n&= \\delta(\\omega_{f0}-\\omega_{\\rm BL})\\delta(\\omega_{f0}-\\omega_{\\rm\nBL}).\n\\end{split}\n\\end{equation}\nAs confirmation, in Fig.\\,\\ref{fig3}(b) we show the time-dependent\nproduct of the functions $F_{\\omega_0}^{(1)}(t,-\\omega_{\\rm IR},0)$\nand $F_{\\omega_0}^{(1)}(t,-\\omega_{\\rm BL},\\Delta_{\\rm BL})$ for\ndifferent $\\omega_{f0}$. As stated above, all situations have in\ncommon that the product is zero at the time when the pulse is\nswitched off, representing energy conservation. Finally, the\nfirst-order transitions for $t=T_p\/2$ are given by\n\\begin{equation}\n\\begin{split}\nd^{(1)}_{0\\rightarrow f}(t>T_p\/2)=&-\\frac{1}{2i}\\left[\\mathcal{E}_{\\rm\nIR}\\langle f|D_{\\rm IR}|0\\rangle\\delta(w_{f0}-\\omega_{\\rm IR})\\right.\n\\\\\n&\\left.+ e^{{\\rm i}\\Delta\\omega_{\\rm BL}}\\mathcal{E}_{\\rm BL}\\langle\nf|D_{\\rm BL}|0\\rangle\\delta(w_{f0}-\\omega_{\\rm BL})\\right].\n\\end{split}\n\\end{equation}\nThe energies of the blue and red photons are not sufficiently high to\nreach the continuum and only the two-photon matrix element developed\nin Eq.\\,\\eqref{eq:d2_2} gives insight into the properties of the\nphotoelectron.\n\nAs a consequence, the amplitude $d^{(1)}$ describes the\nphotoexcitation process of the intermediate $f=5p$ and $f=6p$ states.\nHowever, due to the sharp laser pulses the same final state $f$\ncannot be excited by both photons. Hence, in the resulting\nphotoexcitation probabilities $c_f=|d^{(1)}_{0\\rightarrow\nf}(t>T_p\/2)|^2$ the random phase $\\phi=\\Delta\\omega_{\\rm BL}$ does\nnot play a role, which confirms the full-numerical results shown in\nFig.\\,\\ref{fig1}(b).\n\\begin{figure*}[t!]\n\\includegraphics[width=0.95\\textwidth]{Fig4}\n\\caption{(a) Differential cross section (DCS) when the blue laser field\nis off-resonance, thereby effectively closing the second ionization pathway.\nComparison between two theoretical models (PT:red and\npropagation:gray) and experimental results. For the propagation, the\nblue field was detuned by $\\delta\\omega_{\\rm BL}=0.13$\\,eV. Further,\nthe infrared field was slightly detuned by $\\delta\\omega_{\\rm\nIR}=0.04$\\,eV to manipulate the ratio between both two-photon\namplitudes (see discussions in Sec.\\,\\ref{sec:3}). (b) Same as in (a)\nwith the infrared field completely off-resonance, effectively closing\nthe first ionization pathway. The detuning amount is $\\delta\\omega_{\\rm\nIR}=0.14$\\,eV. (c) Both laser fields resonant with associated\npathways. To achieve pathway amplitudes that are comparable with experiment\nthe infrared field was slightly detuned by $\\delta\\omega_{\\rm\nIR}=0.04$\\,eV. The inset shows a polar plot of the DCS. All\ntheoretical curves are scaled by the same factor and are shifted by a\nconstant offset of 0.1 to aid  comparison with the experimental data.\nThe data points are reproduced from Ref.\\cite{pursehousedynamic}.}\n\\label{fig4}\n\\end{figure*}\nIn panel \\ref{fig3}(c) we present the occupation numbers\n$c_{5p}(t)=|\\langle\\phi_{5p}|\\Psi(t)\\rangle|^2$ and\n$c_{6p}(t)=|\\langle\\phi_{6p}|\\Psi(t)\\rangle|^2$ extracted from the\nperturbative treatment (PT) in Eq.\\eqref{eq:d1} and the numerical\nsimulation scheme developed in Sec.\\,\\ref{sec:2}. We find a remarkable\nagreement between the PT results and the full numerical treatment,\ndemonstrating the transition into the \\emph{cw} limit and the\nvalidity of the perturbative treatment of the two-color ionization\nproblem. Due to the detuning of the IR field the $c_{5s}(t)$\nterm decreases at the end of the pulse while the $c_{6p}(t)$ term belongs to\nresonant excitation ($5s\\rightarrow6p$) in the blue laser field. This\nis the reason for the nearly monotonous increase that is seen. We note that the corresponding\nRabi frequencies $\\Omega_{5s-5p}$ and $\\Omega_{5s-6p}$ are very small which means the occupation numbers in\nFig.\\,\\ref{fig3}(c) represent only the first segment of the first Rabi Cycle which can be identified by\nthe characteristic quadratic dependence on the time.\n\n\\section{Interference between both ionization pathways}\n\\label{sec:4}\n\nAs revealed by the two-photon matrix element in\nEq.\\,\\eqref{eq:2PhotonMatrix}, both laser fields act simultaneously\nto produce the same final photoelectron state, and so we have to deal\nwith two transition amplitudes $t_1$ and $t_2$ which are presented by\nthe two terms. As already demonstrated in Sec.\\,\\ref{sec:2} the final\nstate $|f\\rangle$ is mainly described by a superposition of two\nangular channels $\\ell=0$ and $\\ell=2$. Thus, we may write\n\\begin{equation}\n\\begin{split}\nd^{(2)}_{0\\rightarrow\nf}(\\vartheta_{\\pmb{k}})&= t_1(\\vartheta_{\\pmb{k}})+t_2(\\vartheta_{\\pmb{k}}) \\\\\n&=\\underbrace{S_{\\ell=0}^{(t_1)}(\\vartheta_{\\pmb{k}})e^{i\\phi_{\\ell=0}(k_f)}\n+\nS_{\\ell=2}^{(t_1)}(\\vartheta_{\\pmb{k}})e^{i\\phi_{\\ell=2}(k_f)}}_{t_1=|t_1|e^{i\\varphi_1}}\n\\\\\n&\\quad+\\underbrace{S_{\\ell=0}^{(t_2)}(\\vartheta_{\\pmb{k}})e^{i\\phi_{\\ell=0}(k_f)}\n+\nS_{\\ell=2}^{(t_2)}(\\vartheta_{\\pmb{k}})e^{i\\phi_{\\ell=2}(k_f)}}_{t_2=|t_2|e^{i\\varphi_2}}.\n\\end{split}\n\\end{equation}\nThe final state is in the continuum and is defined by\nEq.\\,\\eqref{eq:continuum}. Consequently, we can define\n\\begin{equation}\n\\begin{split}\nS^{(t_i)}_{\\ell_f}(\\vartheta_{\\pmb{k}})=&(-1)^{2+\\ell_f\/2}\\sqrt{1+\\ell_f\/2}\\begin{pmatrix}\\ell_f&1&1\n\\\\ 0&0&0 \\end{pmatrix}\\begin{pmatrix}1&1&0\\\\0&0&0 \\end{pmatrix}\\\\\n&\\times\\sum_n \\frac{\\langle E_f\\ell_f||\\hat{d}||n\\rangle\\langle\nn||\\hat{d}||5s\\rangle}{\\omega_{0n}-\\omega_1} \\\\\n&\\times \\mathcal{E}_{\\rm IR}\\mathcal{E}_{\\rm BL}\nF^{(2)}_{\\omega_{nf},\\omega_{0n}}(t\\rightarrow\\infty,-\\omega_2,-\\omega_1,0,0)\\\\\n&\\times Y_{\\ell_f,0}(\\vartheta_{\\pmb{k}}).\n\\end{split}\n\\end{equation}\nHere without loss of generality we set the time delays $\\Delta_{\\rm IR\/BL}$ to\nzero since we are in the \\emph{cw} limit. For pathway $t_1$, $\\omega_1=\\omega_{\\rm IR}$ and $\\omega_2=\\omega_{\\rm\nBL}$, while for pathway $t_2$ the opposite is required. Further,\n$\\phi_\\ell(k)=\\delta_\\ell(k)-\\ell\\pi\/2$ while the corresponding pathway phases $\\varphi_{\\rm i}$ are already defined\nin Eq.\\,\\eqref{eq:phase}. Note, that $|t_1(\\vartheta)|^2$ and\n$|t_2(\\vartheta)|^2$ define the DCS for the individual pathways while\nthe interference term between both pathways is given by\n\\begin{equation}\n\\begin{split}\n{\\rm DCS}_{\\rm interf.} &=\nt_1(\\vartheta_{\\pmb{k}})t_2^*(\\vartheta_{\\pmb{k}}) +\nt_2(\\vartheta_{\\pmb{k}})t_1^*(\\vartheta_{\\pmb{k}}) \\\\\n&=\\left|t_1(\\vartheta_{\\pmb{k}})+t_2(\\vartheta_{\\pmb{k}})\\right|^2 -\n(\\left|t_1(\\vartheta_{\\pmb{k}})\\right|^2 +\n\\left|t_2(\\vartheta_{\\pmb{k}})\\right|^2)\n\\end{split}\n\\end{equation}\nThe \\emph{phase difference} related to ${\\rm DCS}_{\\rm interf.}$ is given by\n\\begin{equation}\n\\begin{split}\n\\Delta\\varphi_{12}(\\vartheta_{\\pmb{k}})=\\cos^{-1}\\left[\\frac{t_1(\\vartheta_{\\pmb{k}})t_2^*(\\vartheta_{\\pmb{k}})\n+\nt_2(\\vartheta_{\\pmb{k}})t_1^*(\\vartheta_{\\pmb{k}})}{2\\left|t_1(\\vartheta_{\\pmb{k}})\\right|\n\\left|t_2(\\vartheta_{\\pmb{k}})\\right|}\\right].\n\\end{split}\n\\end{equation}\nThe interference term is the result of differences in the ratios\n$S^{(t_1)}_{\\ell=2}\/S^{(t_1)}_{\\ell=0}$ and\n$S^{(t_2)}_{\\ell=2}\/S^{(t_2)}_{\\ell=0}$ between the $s$- and $d$-\npartial waves associated with the individual pathways $t_1$ and\n$t_2$. These ratios depend crucially on the reduced bound-continuum\ndipole matrix elements $\\langle E_f,\\ell||\\hat{d}||5p\\rangle$ and\n$\\langle E_f,\\ell||\\hat{d}||6p\\rangle$ as well as the bound-bound\ndipole matrix elements $\\langle 5p||\\hat{d}||5s\\rangle$ and $\\langle\n6p||\\hat{d}||5s\\rangle$.\n\nThe perturbative treatment of the two-pathway ionization process shares  some parallels with\nthe theoretical description of the recently developed attosecond measurement techniques \\cite{dahlstrom2013theory}. For instance, the occurrence and spectral characteristics of the $2q$th sideband in the RABBITT scheme \\cite{muller2002reconstruction} stem from the interference between two ionization pathways: the absorption of harmonic $H_{2q-1}$ or $H_{2q+1}$ plus absorption or emission of a laser photon with $\\hbar\\omega$. Hence, similarly to the effects studied in this work, the measured intensity of the side band  depends on the phase difference between the quantum paths. In contrast to typical attosecond experiments, we create a bound wave packet upon absorption of the first photon. This case was studied in photoionization of Potassium where the spectral properties of the initially created bound wave packet was used to eliminate the influence of the dipole phase in the angle-integrated photoelectron spectrum making it possible to fully characterize the attosecond pulses \\cite{pabst2016eliminating}. Similar to the investigated experiment by Pursehouse and Murray, the underlying physical principle is the quantum interference of pathways corresponding to ionization from different energy levels. Moreover, a realistic many-body treatment revealed that correlation effects have only a minor influence in Alkali atoms which supports our theoretical treatment in this work.\n\nIn Fig.\\,\\ref{fig4} we present the individual $t_1$ DCS (a), $t_2$\nDCS (b) and the DCS corresponding to the coherent summation $t_1+t_2$\n(c). In all panels we compare the ionization probabilities extracted\nfrom the full numerical and perturbative treatment with experimental\nresults from measurements performed by Pursehouse and Murray\n\\cite{pursehousedynamic}. In the experiment the individual pathway\ncross sections are obtained by the appropriate  detuning of the\nrespective laser fields: To extract $|t_1|^2$\n($5s\\rightarrow5p\\rightarrow E_f$) the blue laser beam is detuned to block\noccupation of the $6p$ state. To obtain the $|t_2|^2$ amplitude\n($5s\\rightarrow6p\\rightarrow E_f$), the infrared laser field is tuned\nto be off-resonant to the $5p$ transition. To obtain the total\namplitude $|t_1+t_2|^2$ both laser pulses are on resonance to the\nrespective $5s\\rightarrow np$ transitions. As explained in\nSec.\\,\\ref{sec:3} the infrared laser field is always slightly detuned by\na fixed $\\delta\\omega_{\\rm IR}$ so that both two-photon\namplitudes $t_1$ and $t_2$ are of the same magnitude.\n\nIn panel \\ref{fig4}(a) we show the DCS of the individual pathway 1.\nFor the full numerical treatment with a number of 70 optical cycles\nwe used\na blue detuning of $\\delta\\omega_{\\rm BL}=0.13$\\,eV while the red\nfield detuning amounts to $\\delta\\omega_{\\rm IR}=0.04$\\,eV. In the\nperturbative treatment ($n_p\\rightarrow\\infty$) we need a much\nsmaller detuning (in the range of meV). Both theoretical models agree\nextraordinary well with the experiment in general, while minor discrepancies can be found\naround the maxima. A possible explanation is the shift of the final\nenergy by nearly 0.17\\,eV in the full-numerical propagation (finite\nnumber of optical cycles) due to the detuning of both fields which\nchanges slightly the ratios $S^{(t_1)}_{\\ell=2}\/S^{(t_1)}_{\\ell=0}$.\nIn comparison with the experiment, the theoretical models predict the\ncorrect shape of the DCS. They underestimate the data around\n$\\vartheta_{\\pmb{k}}=90^\\circ$ and $\\vartheta_{\\pmb{k}}=270^\\circ$,\nhowever agree well at $\\vartheta_{\\pmb{k}}=0^\\circ$ and\n$\\vartheta_{\\pmb{k}}=180^\\circ$ (along the polarization vectors). The\nsame observations apply for the DCS of the second ionization pathway\n$|t_2|^2$ in panel \\ref{fig4}(b). Figure \\ref{fig4}(c) shows the\nphotoionization probability $|t_1+t_2|^2$ when both laser fields are\nset to resonance with the respective $5s\\rightarrow np$ transitions.\nHere, the infrared field is again detuned by a small\n$\\delta\\omega_{\\rm IR}$ so that both amplitudes $t_1$ and $t_2$\nhave the same magnitude. In addition, we show here the result for\nrather short laser pulses with 12 optical cycles.\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig5}\n\\caption{(a) Interference term ${\\rm DCS}_{\\rm interf.}$ for  two\ntheoretical models and experimental data. Parameters are the same as\nin Fig.\\,\\ref{fig4}. (b) Phase difference $\\varphi_{12}$ between both\nionization pathway amplitudes $t_1(\\vartheta_{\\pmb{k}})$ and\n$t_2(\\vartheta_{\\pmb{k}})$. The yellow line presents a fit of the\nexperimental data to the symmetry-adapted function\n$f(\\vartheta_{\\pmb{k}})=\\sum_{n=0}^2\na_{2n}\\cos^{2n}(\\vartheta_{\\pmb{k}})$. The data points are reproduced from Ref.\\cite{pursehousedynamic}.}\n\\label{fig5}\n\\end{figure}\nSurprisingly, the DCS extracted from the short pulse calculations\nagree very well with the smaller maxima around\n$\\vartheta_{\\pmb{k}}=90^\\circ$ and $\\vartheta_{\\pmb{k}}=270^\\circ$.\nHowever, this rather accidental agreement must be considered within the\nexperimental uncertainties.\n\nIn Fig.\\,\\ref{fig5}(a) we present the interference term ${\\rm\nDCS}_{\\rm interf.}$ for the two developed theoretical models and the\nexperimental data. We see that the amplitude of the interference term\nis clearly non-zero and varies from $13\\%$ to $55\\%$ of the\nnormalized signal shown in Fig.\\,\\ref{fig4}(c). In panel\nFig.\\,\\ref{fig5}(b) we present the corresponding phase difference\n$\\varphi_{12}(\\vartheta_{\\pmb{k}})$ between the two-photon transition\npathways $t_1$ and $t_2$. In comparison to all amplitudes, the\nagreement is less satisfactory which can be explained by the relatively large\nuncertainties due to error propagation through the arccos function\n\\cite{pursehousedynamic}. Interestingly the average value of the phase\nshift is accurately reproduced by both calculations. Under these conditions the predicted angular variation is not very\npronounced and ranges from $110^\\circ$ to $122^\\circ$. Surprisingly,\n the models do not agree as well as for the DCS in\nFig.\\,\\ref{fig4}, which points to the extreme sensitivity of the\nquantum phase to small changes in the transition matrix elements. The yellow curve represents a fit of the experimentally\nobtained phase difference to the symmetry-adapted function $\\sum_{n=0}^2 a_{2n}\\cos^{2n}(\\vartheta_{\\pmb{k}})$.\nIt highlights the agreement with the theoretical prediction with respect to the general shape.\n\nIn the next section we will present mechanisms to manipulate, decrease and\nincrease this modulation.\n\n\\section{Manipulation of the quantum interference}\n\\subsection{Role of the energy gap}\n\\label{sec:5}\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig6}\n\\caption{Interference cross section (a) and phase difference (b)\n$\\varphi_{12}(\\vartheta_{\\pmb{k}})$ for different intermediate state\npairs extracted from the two-photon matrix element in\nEq.\\,\\eqref{eq:2PhotonMatrix}. The IR field is detuned to reach\nequipollent pathway strengths.\nThe numbers in panel (a) represent the ratio\n$\\nu=(d^{(1)}_2\/d^{(1)}_0):(d^{(2)}_2\/d^{(2)}_0)$.}\n\\label{fig6}\n\\end{figure}\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{Fig7}\n\\caption{(a) Addition of a third laser pulse (orange) with amplitude\n$\\mathcal{E}_3$ creates an additional third (direct) ionization pathway $t_{\\rm\ncontr}$ to the final energy $E_{f}$. The blue and IR pulses are the same as\nin Fig.\\,\\ref{fig0}. (b) Interference term ${\\rm DCS}_{\\rm interf}$\nobtained from Eq.\\,\\eqref{eq:interference_mod} for different\nstrengths of the amplitude $t_{\\rm contr}$ relative to\n$t_1+t_2$. (c) Angular dependence of the associated interference\nphase $\\varphi_{12}$ between $t_1$ and $t_2$. All results are\nobtained by numerically calculation with pulse lengths of 75 optical\ncycles while the infrared and blue laser fields have the same field\namplitudes and detunings as in the previous sections.}\n\\label{fig7}\n\\end{figure*}\nIn Fig.\\,\\ref{fig6} we present interference studies on different\nintermediate state pairs defining the two-color ionization amplitudes\n$t_1$ and $t_2$. The blue and the red curves present the interference\nbetween $5p\/7p$ ($\\Delta E= 1.88$\\,eV) and $5p\/8p$ ($\\Delta E=\n2.13$\\,eV) states which have an increasing energy gap $\\Delta E$ between them. The quantity $\\nu=(d^{(t_1)}_{\\ell=2}\/d^{(t_1)}_{\\ell=0})\/\n(d^{(t_2)}_{\\ell=2}\/d^{(t_2)}_{\\ell=0})$ represent the ratios of the\nbound-continuum reduced radial matrix elements between both pathways.\nAs expected for higher Rydberg states this ratio converges quickly to\nthe same number, i.e. the coupling of the $7p$ and $8p$ state to the\ncontinuum is similar. The interference for both state pairs is\nhence comparable. Note that the final energy\n$E_f=E_{5s}+\\hbar\\omega_1 + \\hbar\\omega_2$ is larger in comparison to\nthe $5p\/6p$ case (0.84\\,eV and 1.09\\,eV). Interestingly, ${\\rm\nDCS}_{\\rm interf.}$ has a different shape and changes its sign in\nboth cases, which has an impact on the angular modulation of the phase\ndifference $\\varphi_{12}$. Surprisingly while the amplitude is\nsmaller in comparison with the $5p\/6p$ case shown in\nFig.\\,\\ref{fig5}, the angular variation of the phase is significantly more\npronounced, ranging from $80^\\circ$ to $100^\\circ$. This highlights\nthe change of sign of the interference term (negative sign of the\nargument of the arccos function means a phase larger than\n$90^\\circ$).\n\nThe green and orange curves represent cases when we decrease the\nenergy gap. We chose the intermediate state pairs $6p\/7p$ ($\\Delta\nE=0.51$\\,eV, $E_f=2.22$\\,eV) and $7p\/8p$ ($\\Delta E=0.25$\\,eV,\n$E_f=2.97$\\,eV). Intriguingly, the interference cross section remain\nunaffected when changing to higher lying state pairs. The amplitudes\nof ${\\rm DCS}_{\\rm interf.}$ ranges in both cases from 5\\% to\n50\\% and is negative. However, the angular modulation of the phase\ndifference $\\varphi_{12}$ decreases drastically. We address this\ndevelopment with the dipole matrix element ratio $\\nu$ which\nconverges rapidly to 1, meaning the ratio between the $\\ell=0$\nand $\\ell=2$ angular channels is  nearly the same for both\nionization pathways $t_1$ and $t_2$. According to\nEq.\\,\\eqref{eq:phase} the individual phase shapes are then\ncomparable.\n\nFrom these results we learn that the angular modulation of the phase\ndifference depends critically on the quantity $\\nu$, while the overall\namplitude of the interference term (${\\rm DCS}_{\\rm interf.}$) is\nmore robust and reveals a dependence on the energy gap between the\nintermediate state pair defining $t_1$ and $t_2$.\n\n\\subsection{laser-driven perturbation of ionization pathways}\n\nAnother method for dynamic control of the interference phenomena is\nthe addition of a third control pulse. As represented in the scheme\nin Fig.\\,\\ref{fig7}(a) the corresponding parameters are chosen in a way\nthat  initiates a weak one-photon process directly into the\ncontinuum, so that $\\hbar\\omega_{\\rm contr.}=(E_f-E_{5s})\/\\hbar$.\nThe field amplitude $\\mathcal{E}_{\\rm contr}$ is hence chosen in a\nway that the corresponding transition amplitude $t_{\\rm contr}$ is of\nthe same magnitude as that of the two-color pathways $t_1$\nand $t_2$. As indicated in the modified scheme in\nFig.\\,\\ref{fig7}(a), in the presence of all fields the total\namplitude is given by the coherent sum $t_{\\rm all}=t_1+t_2+t_{\\rm\ncontr.}$. To access the desired interference term ${\\rm\nDCS}_{\\rm interf}=t_1t_2^* + t_1^*t_2$ one has to then perform four\ndifferent measurements: (i) with all fields on resonance to the\nintermediate and final states respectively, (ii) with the blue\nfield to the $6p$ state off-resonance, (iii) with the red field\nto the $5p$ state off-resonance, and (iv) with both red and blue\nfields off-resonant (thereby blocking both\npathways $t_1$ and $t_2$). The interference term is then given by\n\\begin{equation}\n{\\rm DCS}_{\\rm interf}=|t_{\\rm all}|^2 - (|t_{\\rm (ii)}|^2 + |t_{\\rm\n(iii)}|^2 - |t_{\\rm (iv)}|^2)\n\\label{eq:interference_mod}\n\\end{equation}\nwith $t_{\\rm (ii)}=t_1 + t_{\\rm contr}$, $t_{\\rm (iii)}=t_2 + t_{\\rm\ncontr}$ and $t_{\\rm (iv)}=t_{\\rm contr}$.\n\nIn Fig.\\,\\ref{fig7}(b) we present the interference term for\ndifferent strengths of the perturbation by the third (control)\nlaser field. As expected, the additional one-photon ionization route\nhas a large impact on the quantum interference between $t_1$ and $t_2$.\nHere, the field amplitude $\\mathcal{E}_{\\rm contr}$ has to be very\nlow so that the associated $t_{\\rm contr}$ is of the same magnitude\nas the two-photon pathways $t_1$ and $t_2$. In comparison to the\nunperturbed case shown in Fig.\\ref{fig5}(a), the magnitude of the\ninterference is increased in the presence of the control field. In\nstrong contrast to the previous findings, even the sign of the ${\\rm\nDCS}_{\\rm interf}$ can be changed by the effect of the additional\none-photon process when the field amplitude is sufficiently large.\nIt is therefore not surprising that the large impact seen here is directly\ntransferred to the phase $\\varphi_{12}$ associated with the quantum\ninterference. As shown in Fig.\\,\\ref{fig7}(c), the angular variation is\ndrastically increased due to the action of the controlling field. In the\noriginal experiment and theoretical treatment the modulation in the\npolar angle was smaller than 20$^\\circ$. Now we obtain strongly\npronounced phase peaks and a rather complex angular structure of\n$\\varphi_{12}$ with a variation covering more than 140$^\\circ$.\n\nThe addition of the third laser field helps to emphasize that the\ninterference effect is not robust to statistical fluctuations, but is\nunique for every set of laser parameters.\nFor this purpose we slightly detuned the blue laser field so that\nthe $6p$ state was not excited [cf.\\,Fig\\,\\ref{fig8}(a)]. The\nresulting interference phenomenon in the ionization channel then stems\nfrom the superposition of the two-photon pathway 1 (via $5p$\nphotoexcitation) and the one-photon direct photoionization amplitude\nmediated by $E_{\\rm contr.}$. By tuning the parameters of the third field so\nthat the transition strength of the amplitude $t_{\\rm contr.}$ is\nequal to $t_1$ we obtain a characteristic interference as shown by\nthe dark blue curve in Fig.\\,\\ref{fig8}(b). We then varied the\namplitude of the control field in a way that $t_{\\rm\ncontr.}\\in[-1.0t_1,1.0t_1]$ by use of a random number generator. The\nadditional curves in the figure show a statistical average based on the total number of\nrandom amplitudes input to the model. One can clearly see a trend that increasing the\nnumber of random events decreases the interference effect. This shows\nthat for an infinite number of measurements the resulting\ninterference would disappear.\n\\begin{figure}[t]\n\\includegraphics[width=0.95\\columnwidth]{Fig8}\n\\caption{Stochastic nature of the interference effect. In this example one of\nthe ionization pathways is closed by slight detuning of the laser\nfield. We hence\nfind interference between the two-photon amplitude (pathway 1) and\nthe one-photon transition initiated by the third control field\n$E_3(t)$ (scheme panel a). (b) The stochastic nature of the interference\neffect: the amplitude of the one-photon process is randomly varied between\n$-1.0t_1$ and $+1.0t_1$. The curves show the dependence on the number\nof random amplitudes with the reference to $t_{\\rm contr.}=1.0t_1$\n(blue line). Numbers in the round brackets denote the number of\namplitudes used for the statistical average (see text for details).}\n\\label{fig8}\n\\end{figure}\n\n\\section{Conclusions}\n\nIn this paper we have presented a theoretical investigation of\nexperimental interference studies in a single Rubidium atom.\nWe have systematically demonstrated the transition from the short-pulse into\nthe continuous wave regime, and the evolution of the occupation numbers,\nphotoionization probability and quantum phase under an increase of\nthe pulse lengths. We find that for pulse lengths of more than ten\noptical cycles the theoretical description of the ionization scheme\nvia the two-photon matrix element in the frequency regime is\nsufficient, and that this provides all the physical information\nrequired for interference studies.\n\nOur theoretical model provides generally good agreement with the\nexperimental data and predicts a pronounced interference amplitude\n${\\rm DCS}_{\\rm interf.}$ while the angular variation of the\nassociated phase difference $\\varphi_{12}$ is relatively weak. In this\ntreatment we have developed various strategies to manipulate the quantum\ninterference between both photoionization pathways $t_1$ and $t_2$.\nAs an example, we can change the populations of the intermediate\nstates by laser detuning which introduces an imbalance between both\npathways so as to change the interference phenomena. Further,\nby choosing different state pairs $n_1p$ and $n_2p$ we change the\nenergy difference and coupling to the continuum, which again markedly\nchanges the quantum interference. A new method which does not\nchange the intermediate state pairs and the parameters of the blue\nand infrared laser fields is the addition of a third control laser\nfield which perturbs the original transition pathways $t_1$ and $t_2$. An\nappropriate tuning of the corresponding one-photon transition\namplitude into the continuum can even invert the sign of the\ninterference amplitude, as well as produce a much more pronounced\nangular variation of the interference phase. There is no analogy to\nthis third control in the conventional double slit experiments.\n\nAs well as the addition of a third pulse, there are several other\npossibilities to explore the behaviour of the interference phenomenon.\nAs an example, one can show that in Alkali atoms quadrupole\ntransitions into the continuum reveal Cooper minima at kinetic\nenergies below 1eV. Thus, one can choose intermediate state pairs\nwhich lead to final energies in the region of such Cooper minima\nwhile such quadrupole transitions at low intensity are generally\naccessible by structured light fields \\cite{schmiegelow2016transfer}.\nFurther work will hence be dedicated to studying these two-pathway interference\neffects with inhomogeneous light-induced quadrupole transitions.\n\n\\section*{Acknowledgements}\n\nThis work was partially supported by the DFG through SPP 1840 and SFB TRR 227. The EPSRC\nU.K. is acknowledged for current funding through Grant No. R120272.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe Dark Energy Survey (DES) is a next generation\ndeep, wide, multi-band imaging survey that will map 5000 square degrees \nof sky in 525 nights over five years beginning in late 2012.  DES relies on the new \n570-Megapixel, 2.2 degree field-of-view \nDark Energy Camera, known as DECam \\citep{decam}, now installed at the prime focus\nof the 4 m Blanco telescope at the Cerro Tololo Inter-American Observatory (CTIO).  \n\nModern imaging surveys require highly accurate and precise photometry to meet stringent requirements placed on the data \nby the science goals.\nThe requirement for accurate and precision photometry in turn requires accurate and precise calibration of the data products.  \nFor example, DES has a goal of achieving photometry accurate to 1\\% (0.01 mag)\nover the entire survey region.  \nIn order to make measurements with this level of accuracy, care must be taken to \ncalibrate the data appropriately and to remove any spurious effects due to \nweather, the earth's atmosphere, noise and other features of \nthe detector system, and changes in instrumental throughput or response.\nThese calibrations must be made in addition to standard astronomical photometric corrections \nsuch as photometric zeropoint corrections and airmass and color terms.\n\nDECal is a complete calibration system for DECam that will \nallow for the correction of two of these sources of error.  DECal will\nprovide daily flat fields for DECam to correct the pixel-to-pixel variations across the \nCCD detectors; it will also measure the relative throughput\nof the complete telescope+instrument system as a function of wavelength. \nThis spectrophotometric calibration system will allow for the monitoring of the \ninstrumental throughput (e.g., to determine whether optical coatings evolve with time); \nit will also provide accurate knowledge of the filter transmission functions\nthat can be used to calculate precise Supernova k-corrections and improved photometric redshift measurements. \nWe plan to use DECal to monitor the throughput of the telescope at regular\nintervals (i.e., $\\sim$once per month) during the five-year survey to monitor the\ninstrumental performance.\nDECal will be installed permanently at the Blanco telescope and will be available for general use. \n\nSimilar spectrophotometric calibration systems have been tested and\ninstalled on the CTIO 4 m Blanco telescope \\citep{stubbs2007} \nand the PanSTARRS telescope \\citep{panstarrs}.\nBoth of these systems use a tunable laser as a light source. The DECal system uses a \nmonochromator-based light source in place of a tunable laser, which \ngenerally requires less maintenance and other personnel attention than a tunable \nlaser, and is well suited for routine use over an extended period of time. \nThe DECal system also has the advantage of its extended and continuous \nwavelength coverage.  A system such as DECal can operate from the far UV into the infrared (250 $<\\lambda<$ 2400 nm) \nwith relatively minor modifications.  The disadvantage \nof a monochromator-based light source is that it cannot produce as much light\nas a tunable laser, so integration times must be longer and the dome must be kept very dark.  \n\nWe have successfully deployed a prototype of the DECal spectrophotometric calibration system at the Swope \n1 m and du Pont 2.5 m telescopes at Las Campanas Observatory in Chile. \nWe measured the throughput of the $u$, $g$, $r$, $i$, $B$, $V$, $Y$, $J$, $H$, and $K_s$ filters used in the WIRC and RetroCam instruments\nduring the Carnegie Supernova Project with an accuracy of \n1\\% \\citep{stritz}.\nA forthcoming paper on the completed DECal system will present the results of the calibration \nof the DECam instrument using the final DECal system once both DECam and DECal are installed at the CTIO 4m telescope.  \n\nThe complete DECal system consists of a new \nflat field screen, a daily dome flat field illumination system, and \nthe spectrophotometric calibration system.  These systems are described in detail below.\nA schematic of the entire calibration system is shown in Figure \\ref{fig:overview}.\nMore details on the final system may be found in a recent paper by \\citet{decal2012}.\n\n\n\\begin{figure}\n\\plotone{marshall_j_1.eps}\n\\caption{Schematic drawing of the DECal system.\n\\label{fig:overview}\n}\n\\end{figure}\n\n\n\\section{Flat Field Screen}\n\nThe flat field screen is an important part of any dome-based calibration\nsystem and should be carefully considered when designing the calibration scheme.  \nAn ideal flatfield screen would have 100\\% reflectivity at all wavelengths \nof interest and would reflect light only within the acceptance angle of the \ntelescope optics. Since the latter condition is almost impossible to satisfy, \nwe have chosen to provide uniform illumination of the top end of the telescope\nby producing an illumination pattern with a Lambertian profile.  \nA Lambertian surface reflects incident light such that  \nthe surface luminance of the screen as seen by the telescope is isotropic.   A flat field screen with Lambertian scattering\nproperties ensures that all points on the screen illuminate the focal plane\nwith the same angular profile regardless of where the illumination source is \nplaced. If the telescope is well baffled, illumination of this type should be \nequivalent to illuminating the telescope with incident light at only the acceptance angle of the telescope.\n\nThe DECal flat field screen is made of a 2x4 grid of 4 foot by 8 foot \nlightweight aluminum honeycomb panels coated with \na white, highly reflective, almost perfectly Lambertian coating.   \nThese panels are mounted on an extruded aluminum structure using screws that have \nbeen coated with the same coating as the screen.  \nThe white circular portion of the screen is slightly oversized (4.64 m) and is surrounded by a black ring made of \nsheet metal painted with flat black spray paint by Maaco (note that we do not recommend this \nvendor for black coatings).  Our studies show that it is important to \ninclude this outer ring on the flat field screen to minimize stray light reflected into the beam by the \nshiny internal surface of the dome.  We have also determined that any stray light \nreflected by the seams between the two panels or the heads of the screws is minimal \nand has almost no effect ($<<$1\\%) on the precision of photometric measurements that \nhave been flattened with this system; these results will be presented in a forthcoming paper describing the \ndeployed DECal system.\n\nSeveral candidate flat field screen coatings were tested for absolute reflectivity as well as reflectivity \nas a function of incidence angle (Lambertian-ness).\nMore details of this study are provided by \\citet{decal2010}.  \nOut of several candidate screen coatings we finally selected the nearly ideal \n``Duraflect'' coating provided by Labsphere, which provides high reflectivity \n(roughly 95\\% from 350 to 1200nm and greater than 85\\%  for 300 $<\\lambda<$ 2200nm).\nThis coating also boasts a nearly perfectly Lambertian scattering surface.  It is relatively durable and can be \n(gently) cleaned, an important quality for the dome environment.\n\n\n\\section{Dome Flat Field Illumination}\n\nThe DECal daily flat field calibration system provides for daily flat fields to be taken at \nthe telescope.  It uses high-power light-emitting diodes (LEDs) as the illumination source.  \nWe have selected one LED to illuminate each filter bandpass used in the DES survey,\nusually centered on the filter bandpass.   We have also provided LEDs at a wavelength that can \nilluminate a planned u-band filter, which will be added to DECam by CTIO in the near future. Figure \\ref{fig:led}\nshows the spectra of the selected LEDs along with the DES filter bandpasses.\n\n\\begin{figure}\n\\plotone{marshall_j_2.eps}\n\\caption{Spectra of selected DECal LEDs overplotted with the DES $grizy$ (and $u$) filter response curves. The normalization is arbitrary.\n\\label{fig:led}\n}\n\\end{figure}\n\nThe seven LEDs selected to illuminate the DECam filter bandpasses are manufactured by Roithner Lasertechnik and Luxeon Star.  \nThe model numbers of the LEDs are given in Table \\ref{table:leds}.\nThey are positioned at four locations around the top of the telescope ring and provide adequately flat illumination \nof the flat field screen.  \nThe flat field screen does not need to be perfectly uniformly illuminated to produce \na reasonably uniform illumination of the focal plane; however, large scale gradients \nshould be avoided to minimize the need to remove such trends from the data.  \nThe calibration system \ncan be tuned so that the relative power in each bandpass is about the same, i.e., DECam \nexposure times will be the same for each filter.\n\n\\begin{table}[!ht]\n\\label{table:leds}\n\\caption{DECal LEDs}\n\\smallskip\n\\begin{center}\n{\\small\n\\begin{tabular}{lll}\n\\tableline\n\\noalign{\\smallskip}\nManufacturer & Model Number & Central Wavelength\\\\\n\\noalign{\\smallskip}\n\\tableline\n\\noalign{\\smallskip}\nRoithner & H2A1-H365-S & 365 nm\\\\\nRoithner & H2A1-H650 & 650 nm\\\\\nRoithner & H2A1-H780 & 780 nm\\\\\nRoithner & H2A1-H905 & 905 nm\\\\\nRoithner & H2A1-H970 & 970 nm\\\\\nRoithner & H2A1-H1030 & 1030 nm\\\\\nLuxeon Star & Warm white (3100k) Rebel LED & broad\\\\\n\\noalign{\\smallskip}\n\\tableline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\nTesting of similar LED-based dome flat field systems \\citep{ff2005} shows that it \nis not necessary to illuminate the entire filter bandpass to provide adequate \nflat fields.  The exception to this, of course, is in the case that the detector\nquantum efficiency changes rapidly with wavelength.  This is usually not \nthe case for modern CCD detectors.  \n\n\n\\section{Spectrophotometric Calibration System}\n\nIn addition to broadband flat fields, DECal also projects nearly monochromatic (1--10nm bandwidth)\nflat field light onto the flat field screen to be used to measure the relative throughput of the instrument \nas a function of wavelength.  This spectrophotometric calibration is accomplished by \nimaging the monochromatic light incident on the flat field screen with DECam while at the same time\nmonitoring the amount of light on the screen with calibrated photodiodes placed around the top ring of the telescope.  \nThe signal received by the photodiodes is proportional to the light received by DECam; note however that we do not attempt to \nestimate the total amount of light gathered by the telescope, i.e., \nthis is a relative calibration, not an absolute measure of instrumental throughput.\nThe DECam images are compared with the photodiode output to determine the \nrelative sensitivity of the entire telescope+instrument optical system as \na function of wavelength.  \nThis part of the DECal system will be used to spectrophotometrically calibrate \nthe DES survey data, and to monitor the DECam instrument's sensitivity as a function of time.\n\nWe calculate that this system will have a peak output power of 2 mW resulting in approximately \n800 photons\/second\/pixel when imaged by DECam.  This flux level will require exposure times with DECam of \napproximately one minute to obtain adequate signal-to-noise ratio (S\/N) measurements\nof the DECam focal plane; given the relatively bright environment of the Blanco dome we anticipate that \nthe DECal measurements will best be made during a cloudy night: to minimize scattered light from the Sun during the daytime and \nto avoid interfering with regular nighttime observations.\n\n\\subsection{Monochromator}\nThe monochromatic light is produced by a fully automated Czerny-Turner monochromator \n(manufactured by Horiba, model iHR-320). We adjust the \nbandwidth of the light by varying the input slit width. The output slit width is fixed to \n900 microns (the width of the linear fiber bundle that feeds light to the top of the telescope). \nThe monochromator holds two light sources, multiple gratings on a turret, and all of the necessary \norder sorting filters to properly produce the monochromatic light at the appropriate wavelength. \nIt is completely remotely controlled by a Labview interface developed in our lab and is highly flexible \nin producing light of the appropriate wavelength, bandwidth, and interval between requested wavelengths. \n\n\\subsection{Fiber bundle}\nWe have designed and manufactured a custom 75 m long ``line-to-spot'' fiber bundle (assembled by Fibertech Optica) that \nuses a broad-spectrum fiber (Polymicro FBP-300660710). This fiber has excellent \ntransmission both in the UV and the IR, in contrast with standard optical fibers that have either good \ntransmission  in the infrared (due to low OH$^-$ content) or in the UV \n(high OH$^-$ content) but not at both ends of the spectrum simultaneously. \n\nThe bundle consists of 87 fibers with 300 micron cores. At the input end, the fibers are \narranged in 3 parallel lines of 29 fibers each.  The bundle brings the light from \nthe calibration room located beneath the telescope main floor through the telescope cable wrap and\nto the telescope top ring. At a \nlength of 65 m, the fiber bundle splits into four branches, each 10 m long and containing \n21 fibers. These 21 fibers are placed in a compact circular arrangement and coupled to the\nprojection system. Each of the four branches sample the monochromator slit evenly to ensure \nthat they all have the same intensity and spectral content.\n\nA fifth branch bifurcates from the main bundle 1 m from the input slit. It contains the \nthree central fibers and is fed to the monitor spectrometer (see below). To ensure that \nany absorption present in the fiber does not change the measured wavelength profile, \nthis fifth branch is also 75 m long even though the spectrometer is located only 1 m from the monochromator. \n\n\\subsection{Projection system}\n\nTo ensure a uniform \nillumination of the focal plane area, the output from the four fiber ends is collimated \nand then an engineered diffuser (manufactured by RPC Photonics)\nis used to diffuse the light projected onto the flat field screen by the fibers. \nThis type of diffuser can be designed \nto distribute incident collimated light in almost any pattern; in this case \nwe chose a 20 degree angle cone diffuser that projects the light in a top hat shape with a \ndiverging half-angle of 20 degrees, with more than 80\\% of the \nlight exiting the fiber bundle falling within this 40 degree cone of light. Engineered diffusers such as the one used in DECal are a far \nsuperior way to diffuse incident light as compared to a standard ground glass diffuser that would also produce an adequately\ndiffuse light pattern but would scatter much of the light away from the screen. \n\n\\subsection{Spectrometer}\n\nTo monitor in real time the spectral content of the illumination source, \na branch of three fibers bifurcates from the main fiber bundle and is fed to an echelle spectrometer \n(manufactured by Optomechanics Research, model SE-100). \nThe spectrometer measures both the central wavelength \nand the full width at half maximum (FWHM) of the light. For each DECal exposure, a spectrum of the DECal light is obtained and automatically analyzed \nto measure the central wavelength and FWHM of the illumination light with a precision of 0.1 nm. \nThe spectrometer calibration is verified with a Mercury calibration lamp.\n\n\\subsection{Reference photodiodes}\n\nThe goal of the spectrophotometric calibration system is to measure the relative throughput of the \ntelescope+instrument optical system at each wavelength.  This is accomplished by imaging the \nmonochromatic light incident on the flat field screen with the DECam imager while at the same time \nmonitoring the light on the screen to remove any fluctuations in brightness.  \nThe monitoring of the screen is done with the reference photodiodes.  \n\nThe reference photodiodes are mounted \nin the same locations as the fiber projection units at four points around the top ring of the telescope, facing the\nscreen (see insert on Figure \\ref{fig:overview}).   We use 10 mm diameter \nsilicon photodiodes (manufactured by Hamamatsu, model S2281) to measure the light in the range 300 $<\\lambda<$ 1100nm.\nWe acquire the signal from the photodiodes with a data acquisition unit and record the data with the Labview interface.\n\nThe photodiodes we use exhibit a slight change in sensitivity with temperature \nat the far red end of their sensitivity range. This effect is negligible below 1000nm but can reach a \nsensitivity change of up to 1\\%\/\\deg C at 1100nm. We monitor the photodiode temperature in real time \nand correct for it in the data reduction process. \n\n\n\\section{Results}\n\nWe have used a prototype of the DECal spectrophotometric calibration system to measure the \nsystem throughput as a function of wavelength for the WIRC infrared imager on the 2.5 m du Pont telescope and Retrocam optical and infrared imager on the 1 m \nSwope telescope at Las Campanas Observatory, to improve the calibration of the \nCarnegie Supernova Project (CSP) data obtained on these telescopes.\nA description of the DECal prototype used for these measurements is presented by \\citet{decal2010} and the \nresults of the successful calibration are applied to the optical CSP data and presented by \\citet{stritz}.\nIn brief, the DECal prototype used at the du Pont and Swope telescopes included a new flat field screen for each telescope \nwith the same properties but smaller in size than the flat field screen described above, a monochromator smaller but functionally the same \nas the final DECal monochromator, a custom line-to-spot fiber bundle using the same high throughput fibers used in DECal, \nand a monitor photodiode mounted behind the secondary mirror of each telescope.\nHere we present the results of the optical and infrared measurements made using the DECal prototype.\n\nFigure \\ref{fig:results_optical} shows the scans of \nthe six optical filters used to make CSP photometric measurements on \nthe Swope telescope, as well as the scan we made of the telescope+instrument system with no filter in place.   \nThe figures presented in this section give the relative throughput of the telescope, including losses by the primary and secondary mirrors, \ncorrector plate, filter, dewar window and CCD quantum efficiency.  \n\nThe scan of the CSP Retrocam $u$-band filter was particularly \nuseful in analyzing the CSP photometric data.  The manufacturer of the $u$-band filter did not provide \na complete throughput curve with the filter.  An assumption (which turned out to be not entirely correct)\nwas made about the spectral response of the blue edge of the $u$-band filter in the initial data reduction of CSP data; \nwe were able to measure the true spectral response of the $u$-band filter to thereby improve the CSP photometry with our measurements.\n\nFigures \\ref{fig:results_swopeir} and  \\ref{fig:results_irdupont} show scans of the infrared CSP \nbandpasses used on the Swope and du Pont telescopes, respectively.  \nEach infrared filter scan has been normalized independently, i.e., the relative throughput from filter to filter is not \nreflected in these figures.\nThese data will also be used to calibrate the CSP infrared photometric measurements, to be published in a forthcoming \nCSP data release.\nThe infrared system scan measurements had slightly lower signal-to-noise (S\/N) than the optical measurements; \nthis is the source of the noise seen at the red end of the $K_s$-band data in Figure \\ref{fig:results_irdupont}.\nHowever, we note that with multiple measurements or increased signal from a brighter light source or even a higher throughput \nmonochromator, higher S\/N measurements of the infrared filters could be made. \nThat is, a system such as DECal could be used to determine the spectral response of an imaging system well into the infrared\nwith only minimal modification.\n\n\\begin{figure}\n\\plotone{marshall_j_3.eps}\n\\caption{DECal prototype scan of the optical filters used by the CSP survey in the Retrocam instrument on the Swope telescope.\n\\label{fig:results_optical}\n}\n\\end{figure}\n\n\n\\begin{figure}\n\\plotone{marshall_j_4.eps}\n\\caption{DECal prototype scan of the infrared filters used by the CSP survey in the Retrocam instrument on the Swope telescope.\n\\label{fig:results_swopeir}\n}\n\\end{figure}\n\n\\begin{figure}\n\\plotone{marshall_j_5.eps}\n\\caption{DECal prototype scan of the infrared filters used by the CSP survey in the WIRC instrument on the du Pont telescope.\n\\label{fig:results_irdupont}\n}\n\\end{figure}\n\n\\section{Summary}\n\nDECal is a new calibration system that provides broadband and monochromatic flat fields for the DECam instrument \non the CTIO Blanco  4 m telescope.  DECal will\nbe used to flatten optical DECam images and to monitor the transmission function of the instrument+telescope system \nand provide spectrophotometric calibration of the system.\n\nWe have fully tested the components of DECal and have deployed a prototype of the system on the Las Campanas Observatory \nSwope and du Pont telescopes.  This prototype has informed the design of the final \nDECal system and was used to successfully calibrate the Carnegie Supernova Project optical and infrared photometry.\n\nThe DECal and DECam systems are currently in the final stages of installation on the CTIO Blanco 4 m telescope and will\nbe commissioned in Fall 2012.  We will use the DECal broadband flat field system to produce daily flat fields \nfor the DECam instrument, and the DECal spectrophotometric calibration system to produce monthly scans of the \nspectral response of the complete telescope+instrument system, through each DES filter and at all wavelengths.  \nThese data will be used to measure the \nspectral response of the instrument, which will enable the highly accurate and precise photometric measurements required by DES.  \nThe DECal system will also be used to monitor any changes in the spectral response of the instrument, \nincluding evolution of filter functions, degradation of optical coatings, and off-band optical leaks in the filters.  \n\n\n\\acknowledgements Texas A\\&M University thanks Charles R. '62 and Judith G. Munnerlyn, \nGeorge P. '40 and Cynthia Woods Mitchell, and their families for support of astronomical \ninstrumentation activities in the Department of Physics and Astronomy.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\\setcounter{equation}{0}\\setcounter{Theorem}{0}\\setcounter{Definition}{0}\n\nThe assumption of global conformal invariance -- which says that we are\ndealing with a single valued  representation of $SU(2,2)$ rather than\nwith a representation of its covering -- in  (4-dimensional) Minkowski\nspace has surprisingly strong consequences \\cite{NT01}. Combined with\nthe Wightman axioms, it implies \\emph{Huygens locality}, which yields\nthe vertex-algebra-type condition\n\\begin{equation}\\label{eq1.1}\n\\bigl(({\\mathrm{x}}-{\\mathrm{y}})^2\\bigr)^n \\, \\bigl[\\phi({\\mathrm{x}}), \\psi({\\mathrm{y}})\\bigr] = 0\n\\quad \\text{for} \\quad n \\gg 0 \\end{equation} for any pair $\\phi, \\psi$ of local\nBose fields ($n\\gg 0$ meaning ``$n$ sufficiently large''). Huygens\nlocality and energy positivity imply, in turn, rationality of\ncorrelation functions. A GCI quantum field theory (QFT) that admits a\nstress-energy tensor (something, we here assume) necessarily involves\ninfinitely many conserved symmetric tensor currents in the operator\nproduct expansion (OPE) of any Wightman field with its conjugate.\nThe twist two contributions give rise to a harmonic {\\it bifield}\n$V({\\mathrm{x}},{\\mathrm{y}})$, which is an important tool in the study of GCI QFT models\n\\cite{NST02,NST03,NRT05,BNRT07,NRT07}.  The spectacular development of\n2-dimensional (2D) \\emph{conformal field theory} in the 1980's\nis based on the preceding study of infinite dimensional (Kac--Moody\nand Virasoro) Lie algebras and their representations. A\nstraightforward generalization of this tool did not seem to apply\nin higher dimensions. After the first attempts\nto construct (4D) Poincar{\\'e} invariant Lie fields led to examples\nviolating energy positivity \\cite{L67}, it was proven \\cite{B76},\nthat scalar Lie fields do not exist in three or more dimensions.\nIt is therefore important to realize that the argument does not pass to\n{\\em bi\\\/}fields, and that the above mentioned harmonic bifields do\ngive rise to infinite dimensional Lie algebras.\n\nConsider bilocal fields of the form\n\\begin{equation} \\label{eq1.1a}\nV_M({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, \\sum_{ij} \\, M_{ij} \\ {:}\\varphi_i({\\mathrm{x}})\\varphi_j({\\mathrm{y}}){:}\n\\,, \\end{equation}\nwhere $M$ is a real matrix and $\\varphi_j$ are a system of independent\nreal massless free fields. According to Wick's theorem, the\ncommutator of $V_{M_1}({\\mathrm{x}}_1,{\\mathrm{x}}_2)$ and $V_{M_2}({\\mathrm{x}}_3,{\\mathrm{x}}_4)$ is:\n\\begin{eqnarray}\n\\label{eq2.1n}\n&& \\hspace{-20pt}\n\\bigl[V_{M_1}({\\mathrm{x}}_1,{\\mathrm{x}}_2), V_{M_2}({\\mathrm{x}}_3,{\\mathrm{x}}_4)\\bigr]\n\\, = \\,\n\\Delta_{2,3} \\, V_{M_1M_2}({\\mathrm{x}}_1,{\\mathrm{x}}_4)\n\\, + \\,\n\\Delta_{2,4} \\, V_{M_1\\trp{M_2}}({\\mathrm{x}}_1,{\\mathrm{x}}_3)\n\\nonumber \\\\ && \\hspace{-20pt}\n\\qquad \\hspace{10pt}\n\\, + \\\n\\Delta_{1,3} \\, V_{\\trp{M_1}M_2}({\\mathrm{x}}_2,{\\mathrm{x}}_4)\n\\, + \\,\n\\Delta_{1,4} \\, V_{\\trp{M_1}\\trp{M_2}}({\\mathrm{x}}_2,{\\mathrm{x}}_3)\n\\nonumber \\\\ && \\hspace{-20pt}\n\\qquad \\hspace{10pt}\n\\, + \\\n\\mathrm{tr} \\, (M_1 M_2) \\Delta_{12,34} + \\mathrm{tr} \\, (\\trp{M_1}\nM_2) \\Delta_{12,43}, \n\\end{eqnarray}\nwhere $\\trp{M}$ is the transposed matrix, $\\Delta_{j,k}$ is the free field\ncommutator, $\\Delta_{j,k}$ $=$ $\\Delta^+_{j,k} - \\Delta^+_{k,j}$,\\ and\n$\\Delta_{jk,lm}$ $=$ $\\Delta^+_{j,m}\\Delta^+_{k,l} - \\Delta^+_{m,j}\\Delta^+_{l,k}$\\\nfor\\ $\\Delta^+_{j,k}$ $:=$ $\\Delta_+({\\mathrm{x}}_j$ $-$ ${\\mathrm{x}}_k)$, the two point\nmassless scalar correlation function.\n\nIt is one of the main results of \\cite{NRT07} that the same abstract\nstructure can be derived from first principles in GCI quantum field theory.\nMore precisely, the twist two bilocal fields appearing in the OPE of\nany two scalar fields of dimension 2 can be linearly labeled by\nmatrices $M$ such that the commutation relations \\eqref{eq2.1n} hold. {}\nFrom this, the representation \\eqref{eq1.1a} can be deduced.\nIn the present paper we shall consider only finite size matrices; in\ngeneral, the system of independent massless free fields can be infinite\nand then the $M$'s should be assumed to be Hilbert--Schmidt operators.\n\nThe question arises, whether there are nontrivial linear subspaces\n$\\mathcal{M}$ of real matrix algebras upon which the commutation\nrelations of the corresponding bifields $V_M$ ($M \\in \\mathcal{M}$) close.\nWe shall call such systems of bifields \\textit{Lie systems},\nor, \\textit{Lie bifields}.\nIt follows from (\\ref{eq2.1n}) that if $\\mathcal{M}$ is a {\\em $t$-subalgebra}\n(i.e., a subalgebra closed under transposition)\nof the real matrix algebra, then $\\{V_M\\}_{M \\, \\in \\, \\mathcal{M}}$\nis a Lie system. Conversely, any Lie system corresponds to a subalgebra\n$\\mathcal{M}$ such that $\\trp{M_1}M_2$, $M_1\\trp{M_2}$,\n$\\trp{M_1}\\trp{M_2}$ $\\in\\mathcal{M}$ whenever $M_1,M_2 \\in\\mathcal{M}$.\nIn particular, if $\\mathcal{M}$ contains the identity matrix,\nthen it is a $t$-subalgebra.\n\n\\section{$t$-subalgebras of real matrix algebras}\\label{sec2a}\n\\setcounter{equation}{0}\\setcounter{Theorem}{0}\\setcounter{Definition}{0}\n\nLet us consider $t$-subalgebras $\\mathcal{M}$ of the matrix algebra $Mat(L,{\\mathbb R})$,\nwhere $L$\nis a positive integer (equal to the number of fields $\\varphi_j$).\nThe classification of all such $\\mathcal{M}$ is a classical mathematical problem,\nwhich goes back to F.G.~Frobenius, I.~Schur, and J.H.M.~Wedderburn\n(see, e.g., \\cite[Chapter XVII]{L02} and \\cite[Chapter 9, Appendix II]{B82}).\n\nWe first observe that $\\mathcal{M}$ is equipped with the Frobenius inner product\n\\begin{equation}\\label{eq2.1xx}\n\\langle M_1, M_2 \\rangle = \\mathrm{tr} \\, (\\trp{M_1}M_2) = \\sum_{ij} \\,\n(M_1)_{ij} \\, (M_2)_{ij} \\,,\n\\end{equation}%\nwhich is symmetric, positive definite, and has the property\n$\\langle M_1 M_2,\\,M_3 \\rangle=\\langle M_1,$ $M_3\\;\\trp{M_2} \\rangle$.\nThis implies that for every right ideal $\\mathcal{I} \\subset\\mathcal{M}$,\nthe orthogonal complement $\\mathcal{I}^\\perp$ is again a right ideal.\nNote also that $\\mathcal{I}$ is a right ideal if and only if\n$\\trp{\\mathcal{I}}$ is a left ideal.\nTherefore, $\\mathcal{M}$ is a {\\em semisimple} algebra (i.e., a direct sum of\nleft ideals), and every module over $\\mathcal{M}$ is a direct sum of\nirreducible ones.\n\nNow assume, without loss of generality, that the algebra\n$\\mathcal{M} \\subset End_{\\,{\\mathbb R}}\\ \\mathcal{L} \\cong Mat(L,{\\mathbb R})$\nacts irreducibly on the vector space $\\mathcal{L} \\cong {\\mathbb R}^L$.\nLet $\\mathcal{M}'\\subset End_{\\,{\\mathbb R}}\\ \\mathcal{L}$ be the {\\em commutant} of $\\mathcal{M}$,\ni.e., the set of all matrices $M$ commuting with all elements of $\\mathcal{M}$.\nThen by Schur's lemma\n(whose real version \\cite{L02} is much less popular than the complex one),\n$\\mathcal{M}'$ is a real division algebra.\nBy the Frobenius theorem, $\\mathcal{M}'$ is isomorphic to ${\\mathbb R}$, ${\\mathbb C}$, or ${\\mathbb H}$\nas a real algebra (where ${\\mathbb H}$ denotes the algebra of {\\em quaternions}).\nFinally, the classical Wedderburn theorem gives that\n$\\mathcal{M}$ is isomorphic to the matrix algebra $End_{\\mathcal{M}'}\\mathcal{L}$.\nIn addition, since $\\mathcal{M}$ is closed under transposition, then $\\mathcal{M}'$\nis also a $t$-algebra, and the transposition in $\\mathcal{M}'$\ncoincides with the conjugation in ${\\mathbb R}$, ${\\mathbb C}$, or ${\\mathbb H}$, respectively.\n\nObserve that, since $\\mathcal{M} \\cong End_{\\mathbb F}\\,\\mathcal{L}$\n(where ${\\mathbb F}={\\mathbb R}$, ${\\mathbb C}$, or ${\\mathbb H}$),\nwe can view $\\mathcal{L}$ as a left ${\\mathbb F}$-module on which $\\mathcal{M}$ acts\n${\\mathbb F}$-linearly. Alternatively, $\\mathcal{L}$ can be made an $(\\mathcal{M},{\\mathbb F})$-bimodule\nby setting $M \\cdot f \\cdot M' := M \\, (\\trp{M'}) \\, f$ for\n$f \\in \\mathcal{L}$, $M \\in \\mathcal{M}$ and $M' \\in \\mathcal{M}' \\cong {\\mathbb F}$.\nThen the embedding ${\\mathbb F}\\subset End_{\\mathbb F}\\,\\mathcal{L}\\cong\\mathcal{M}$ endows $\\mathcal{L}$\nwith the structure of an ${\\mathbb F}$-bimodule. In other words,\nwe have two commuting copies,\nleft and right, of ${\\mathbb F}$ in $End_{\\,{\\mathbb R}}\\ \\mathcal{L}$,\nwhich are subalgebras of $\\mathcal{M}$ and $\\mathcal{M}'$, respectively.\nMoreover, denoting $N=dim_{{\\mathbb F}} \\, \\mathcal{L}$, we have:\n$L = dim_{{\\mathbb R}} \\, \\mathcal{L} = N \\, dim_{{\\mathbb R}} \\, {\\mathbb F} =N,2N$ or $4N$\nwhen ${\\mathbb F}={\\mathbb R},{\\mathbb C}$ or ${\\mathbb H}$, respectively.\n\nIf $\\mathcal{M}$ is not an irreducible $t$-subalgebra of $Mat(L,{\\mathbb R})$,\ni.e., $\\mathcal{L} \\cong {\\mathbb R}^L$ is not an irreducible $\\mathcal{M}$-module,\nthen $\\mathcal{L}$ splits into irreducible submodules, each of them of the above\nthree types:\n\\begin{equation}\\label{e1.3}\n\\mathcal{L} \\, = \\, \\mathcal{L}_{{\\mathbb R}} \\, \\oplus \\, \\mathcal{L}_{{\\mathbb C}} \\, \\oplus \\, \\mathcal{L}_{{\\mathbb H}} \\,,\n\\end{equation} where each $\\mathcal{L}_{{\\mathbb F}}$ (${\\mathbb F}={\\mathbb R},{\\mathbb C},{\\mathbb H}$)\nis an ${\\mathbb F}$-module such that $\\mathcal{M}$ acts on it ${\\mathbb F}$-linearly.\n In our QFT application, the space $\\mathcal{L}$ is the real linear span of the\nreal massless scalar fields $\\varphi_j$, and then a Lie system of bifields\n$V_{M}$ splits into three subsystems: of types ${\\mathbb R}$, ${\\mathbb C}$ and ${\\mathbb H}$.\nThe first two cases were considered in a previous paper \\cite{BNRT07}\nand led to gauge groups of type $U(N,{\\mathbb R}) = O(N)$ and $U(N,{\\mathbb C}) = U(N)$, respectively,\nwhere $N=dim_{{\\mathbb F}} \\, \\mathcal{L}$.\nHere we are going to consider the third case in which, as we shall see,\nthe gauge groups that arise are of type $U(N,{\\mathbb H}) = Sp(2N)$,\nthe compact real form of the {\\em symplectic group}.\n\nIn each of the three cases, the associated infinite-dimensional\nLie algebra (\\ref{eq2.1n}) has a central charge proportional to the\norder $N$ of the gauge group $U(N,{\\mathbb F})$.\n\n\n\\section{Irreducible Lie bifields and associated dual pairs}\\label{sec2}\n\\setcounter{equation}{0}\\setcounter{Theorem}{0}\\setcounter{Definition}{0}\n\nIn this section we consider Lie bifields $\\{V_{M}\\}_{M\\in\\mathcal{M}}$\ncorresponding to irreducible $t$-subalgebras $\\mathcal{M}$ of $Mat(L,{\\mathbb R})$.\nAs discussed in the previous section, we have\n$\\mathcal{M}\\cong End_{\\mathcal{M}'}\\mathcal{L}$, where $\\mathcal{L} \\cong {\\mathbb R}^L$ and\nthe commutant $\\mathcal{M}'\\cong$ ${\\mathbb R}$, ${\\mathbb C}$, or~${\\mathbb H}$.\n\nIn the case when $\\mathcal{M}'\\cong{\\mathbb R}$ and $dim_{\\mathbb R}\\, \\mathcal{L}=1$, we have one bifield\n\\begin{equation}\n\\label{eq2.1a}\nV({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, {:}\\varphi({\\mathrm{x}}) \\varphi({\\mathrm{y}}){:} \\,. \\end{equation}\nMore generally, $V$ can be taken a sum of $N$ independent copies\nof Lie bifields of type (\\ref{eq2.1a}),\n\\begin{equation}\n\\label{eq2.2a}\nV({\\mathrm{x}},{\\mathrm{y}}) \\equiv V_{(N)}({\\mathrm{x}},{\\mathrm{y}}) \\, = {:}\\mbf{\\varphi}({\\mathrm{x}})\n\\mbf{\\varphi}({\\mathrm{y}}){:} \\, =  \\mathop{\\sum}\\limits_{j \\, = \\, 1}^N\n{:}\\varphi_j({\\mathrm{x}}) \\varphi_j({\\mathrm{y}}){:} \\,,\n\\end{equation}\nwhich is invariant under the \\textit{gauge group} $O(N)$ (including\nreflections). Here $L=N$ and $O(N)$ is realized as\nthe group of linear automorphisms of $\\mathcal{L} =\nSpan_{{\\mathbb R}} \\{\\varphi_j\\}$ preserving the quadratic form\n(\\ref{eq2.2a}) in $\\varphi_j$. In this case the\n\\textit{field Lie algebra} (i.e., the Lie algebra of field modes\ncorresponding to the eigenvalues of the one-particle energy, see the Appendix)\nis isomorphic to a central extension of $sp(\\infty,{\\mathbb R})$ of central charge $N$; see \\cite{BNRT07}.\n\nThe case when $\\mathcal{M}' \\cong {\\mathbb C}$ and $dim_{\\mathbb C}\\, \\mathcal{L}=1$ is given by two\nreal bifields, $V_{\\Mbf{1}}$ and $V_{\\varepsilon}$ that correspond to the $2\n\\times 2$ matrices\n\\begin{equation}\\label{eq2.3a}\n\\Mbf{1} \\, = \\, \\left(\\hspace{-4pt} \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\hspace{-2pt}\\right)\n\\, , \\quad\n\\varepsilon \\, = \\, \\left(\\hspace{-4pt} \\begin{array}{rc} 0 & 1 \\\\ -1 & 0 \\end{array} \\hspace{-2pt}\\right) .\n\\end{equation}\nThey are thus generated by two independent real massless fields $\\varphi_1 ({\\mathrm{x}})$ and $\\varphi_2({\\mathrm{x}})$:\n\\begin{eqnarray}\\label{eq2.4a}\nV_{\\Mbf{1}} ({\\mathrm{x}},{\\mathrm{y}}) \\, = && \\hspace{-15pt} {:}\\varphi_1({\\mathrm{x}}) \\varphi_1({\\mathrm{y}}){:} + {:}\\varphi_2({\\mathrm{x}}) \\varphi_2({\\mathrm{y}}){:} \\,,\n\\nonumber \\\\\nV_{\\varepsilon} ({\\mathrm{x}},{\\mathrm{y}}) \\, = && \\hspace{-15pt} {:}\\varphi_1({\\mathrm{x}}) \\varphi_2({\\mathrm{y}}){:} - {:}\\varphi_2({\\mathrm{x}}) \\varphi_1({\\mathrm{y}}){:}\n\\,.\n\\end{eqnarray}\nCombining $\\varphi_1$ and $\\varphi_2$ into one complex field\n$\\mbf{\\varphi} ({\\mathrm{x}}) = \\varphi_1 ({\\mathrm{x}}) + i \\varphi_2({\\mathrm{x}})$ we get that\n$V_{\\Mbf{1}}$ and $V_\\varepsilon$ are the real and the imaginary parts of the\ncomplex bifield\n\\begin{equation}\\label{eq2.5a}\nW ({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, {:}\\mbf{\\varphi}^* ({\\mathrm{x}}) \\mbf{\\varphi} ({\\mathrm{y}}){:}\n\\, = \\, V_{\\Mbf{1}} ({\\mathrm{x}},{\\mathrm{y}}) + i \\, V_\\varepsilon ({\\mathrm{x}},{\\mathrm{y}}) .\n\\end{equation}\nTaking again $N$ independent copies of such Lie bifields,\n\\begin{equation}\\label{eq2.6a}\nW_{(N)} ({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, \\mathop{\\sum}\\limits_{j \\, = \\, 1}^N  {:}\\mbf{\\varphi}^*_j ({\\mathrm{x}}) \\mbf{\\varphi}_j ({\\mathrm{y}}){:}\n\\, , \\quad\n\\mbf{\\varphi}_j ({\\mathrm{x}}) \\, = \\, \\varphi_{1,j} ({\\mathrm{x}}) + i \\, \\varphi_{2,j} ({\\mathrm{x}}),\n\\end{equation}\nwe get a {\\em gauge group} $U (N)$, where $L=2N$.\nThe {\\em field Lie algebra} in this second case is\nisomorphic to a central extension of $u(\\infty,\\infty)$ again of central charge $N$ (\\cite{BNRT07}).\n\nFinally, for $\\mathcal{M}' = {\\mathbb H}$ the minimal size of the matrices in\n$\\mathcal{M}$ is four. We can formally derive the basic bifields $V_{M}$ in\nthis case as in the above complex case~(\\ref{eq2.5a}). Let us combine\nthe four independent scalar fields $\\varphi_j ({\\mathrm{x}})$ ($j=0,1,2,3$)\nin a single ``quaternionic-valued'' field and its conjugate:\n\\begin{eqnarray}\n\\label{eq2.6}\n\\mbf{\\varphi} ({\\mathrm{x}}) \\, = && \\hspace{-15pt} \\varphi_0 ({\\mathrm{x}}) + \\varphi_1 ({\\mathrm{x}}) \\, I + \\varphi_2 ({\\mathrm{x}}) \\, J + \\varphi_3 ({\\mathrm{x}}) \\, K\n, \\quad\n\\nonumber \\\\\n\\mbf{\\varphi}^+ ({\\mathrm{x}}) \\, = && \\hspace{-15pt} \\varphi_0 ({\\mathrm{x}}) - \\varphi_1 ({\\mathrm{x}}) \\, I - \\varphi_2 ({\\mathrm{x}}) \\, J - \\varphi_3 ({\\mathrm{x}}) \\, K,\n\\end{eqnarray}\nwhere $I, J, K$ are the (imaginary) quaternionic units\nsatisfying $IJ = K = -JI,\nI^2 = J^2 = K^2 = -1$. This allows us to write a quaternionic bifield $Y$ as\n\\begin{equation}\n\\label{eq2.7}\nY({\\mathrm{x}},{\\mathrm{y}}) \\, = \\, {:}\\mbf{\\varphi}^+({\\mathrm{x}}) \\, \\mbf{\\varphi}({\\mathrm{y}}){:}\n\\, = \\,  V_0({\\mathrm{x}},{\\mathrm{y}}) + V_1({\\mathrm{x}},{\\mathrm{y}}) \\, I + V_2({\\mathrm{x}},{\\mathrm{y}}) \\, J + V_3({\\mathrm{x}},{\\mathrm{y}}) \\, K,\n\\end{equation}\nwhere the components $V_\\alpha$ ($\\alpha = 0, 1, 2, 3$) of $Y$ can be\nfurther expressed in terms of the 4-vectors $\\mbf{\\varphi}$ and a\n$4\\times 4$ matrix realization of the quaternionic units in a manner\nsimilar to~(\\ref{eq2.4a}):\n\\begin{eqnarray}\n\\label{eq2.8}\n&\nV_\\alpha({\\mathrm{x}},{\\mathrm{y}}) \\, \\equiv \\, V_{\\ell_{\\alpha}} \\bigl({\\mathrm{x}},{\\mathrm{y}}\\bigr) \\,\n\\, = \\, {:}\\mbf{\\varphi}(x) \\,\\ell_\\alpha \\, \\Mbf{\\varphi}(y){:} \\, , \\quad\n&\n\\nonumber \\\\\n&\n\\nonumber \\\\\n&\n\\ell_0 = \\Mbf{1} ,\\qquad\n\\ell_1 \\, = \\, \\left(\\hspace{0pt} \\begin{array}{rrrr}                                                                                                                                       0 & 1 & 0 & 0 \\\\\n{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & {\\hspace{-8pt}}-{\\hspace{-2pt}}1 \\\\\n0 & 0 & 1 & 0\n\\end{array}\n\\right)\n,\\qquad\n&\n\\nonumber \\\\\n&\n\\ell_2 \\, = \\,\n\\left(\\hspace{0pt}\n\\begin{array}{rrrr}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 & 0 & 0 \\\\\n0 &{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 & 0\n\\end{array} \\right)\n,\\qquad\n\\ell_3 \\, = \\,\n\\left(\\hspace{0pt}\n\\begin{array}{rrrr}\n0 & 0 & 0 & 1 \\\\\n0 & 0 &{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n{\\hspace{-8pt}}-{\\hspace{-2pt}}1 & 0 & 0 & 0\n\\end{array} \\right) .\n\\end{eqnarray}\nIt is straightforward to check that the $4\\times 4$ matrices $\\ell_{\\alpha}$\ngenerate the quaternionic algebra ${\\mathbb H} \\cong \\mathcal{M}$.\nThe commutant $\\mathcal{M}'$ in $Mat(4,{\\mathbb R})$ is spanned by the unit matrix and\nanother realization of the imaginary quaternionic units as a set of real\nantisymmetric $4\\times 4$ matrices $r_k$ $(k=1,2,3)$.\nThe two sets $\\{r_k\\}_{k\\, = \\, 1}^3$ and $\\{\\ell_k\\}_{k\\, = \\, 1}^3$\ncorrespond to the splitting of the Lie algebra $so(4)$\ninto a direct sum of two $so(3)$ algebras:\n\\begin{eqnarray}\n\\label{eq2.9}\n&\n\\ell_1 \\, = \\, \\sigma_3 \\otimes \\varepsilon\n\\, , \\quad\n\\ell_2 \\, = \\, \\varepsilon \\otimes \\Mbf{1}\n\\, , \\quad\n\\ell_3 \\, = \\, \\ell_1 \\, \\ell_2 \\, = \\, \\sigma_1 \\otimes \\varepsilon \\,,\n\\quad\n&\n\\nonumber \\\\\n&\nr_1 \\, = \\, \\varepsilon \\otimes \\sigma_3\n\\, , \\quad\nr_2 \\, = \\, \\Mbf{1} \\otimes \\varepsilon\n\\, , \\quad\nr_3 = r_1 \\, r_2 \\, = -r_2 \\,r_1 \\, = \\, \\varepsilon \\otimes \\sigma_1  \\,,\n\\quad\n&\n\\end{eqnarray}\nwhere $\\sigma_k$\nare the Pauli matrices and $\\varepsilon = i\\sigma_2$\nas in (\\ref{eq2.3a}).\n\nWe shall demonstrate that the quaternionic field $Y$ (\\ref{eq2.7}) generates\na central extension of\nthe Lie algebra\\footnote{%\nFor a description of the Lie algebra $so^* (2n)$ of the noncompact\ngroup $SO^*(2n)$ \nand of its highest weight representations, see \\cite{EHW83}. For an oscillator realization of the\nLie superalgebra $osp(2m^*\\vert 2n)$ (with even subalgebra $so^*(2m)\\times\nsp(2n)$), see \\cite{GS91}.\nIf we view $so^* (4\\infty)$ as an inductive limit of $so^*(4n)$\nthen the central extension is trivial.}\n$so^* (4\\infty)$.\nTo this end, we represent $Y$ by a pair of complex bifields\n\\begin{equation}\\label{eq3.11it}\n\\begin{array}{l}\nW ({\\mathrm{x}},{\\mathrm{y}}) = \\, \\frac{1}{2} \\, \\Bigl(V_0 ({\\mathrm{x}},{\\mathrm{y}}) + i V_3\n({\\mathrm{x}},{\\mathrm{y}})\\Bigr) \\,\n\\\\\n\\qquad \\qquad \\qquad\n= \\, :{\\hspace{-2pt}} \\psi^*_1 ({\\mathrm{x}}) \\, \\psi_1 ({\\mathrm{y}}) {\\hspace{-2pt}}:\n+ :{\\hspace{-2pt}} \\psi^*_2 ({\\mathrm{x}}) \\, \\psi_2 ({\\mathrm{y}}) {\\hspace{-2pt}}:\n\\, = \\, W ({\\mathrm{y}},{\\mathrm{x}})^*\n\\, , \\quad\n\\\\\n\\hfill\n\\\\\nA ({\\mathrm{x}},{\\mathrm{y}}) = \\, \\frac{1}{2} \\, \\Bigl(V_1 ({\\mathrm{x}},{\\mathrm{y}}) - i V_2({\\mathrm{x}},{\\mathrm{y}})\\Bigr)\n\\\\\n\\qquad \\qquad \\qquad\n= \\, \\psi_1 ({\\mathrm{x}}) \\, \\psi_2 ({\\mathrm{y}}) - \\psi_2 ({\\mathrm{x}}) \\,\n\\psi_1 ({\\mathrm{y}}) \\, = \\, - A ({\\mathrm{y}},{\\mathrm{x}})\n\\, , \\qquad\n\\end{array}\n\\end{equation}\nand their conjugates, where $\\psi_{\\alpha}$ are complex linear\ncombinations of $\\varphi_{\\nu}$:\n\\begin{equation}\\label{eq3.12it}\n\\psi_1 \\, = \\, \\frac1{\\sqrt{2}} \\Bigl( \\varphi_0 + i \\varphi_3 \\Bigr)\n\\, , \\quad\n\\psi_2 \\, = \\, \\frac1{\\sqrt{2}} \\Bigl( \\varphi_1 - i \\varphi_2 \\Bigr)\\,.\n\\end{equation}\n\nSubstituting as above each $\\varphi_{\\nu}$ (respectively $\\psi_{\\alpha}$) by\nan $N$-vector of commuting free fields we can write the nontrivial local\ncommutation relations (CR) of $W(1,2)$ $\\equiv W ({\\mathrm{x}}_1,{\\mathrm{x}}_2)$ and $A(1,2)$\nin the form\n\\begin{eqnarray}\\label{eq3.13it}\n\\bigl[ W^* (1,2), W(3,4) \\bigr]\n\\, = && \\hspace{-15pt}\n\\Delta_{1,3} \\, W(2,4) + \\Delta_{2,4} \\, W^* (1,3) + 2N \\Delta_{12,43} \\, ; \\quad\n\\\\ \\label{eq3.14it}\n\\bigl[ W(1,2), A(3,4) \\bigr]\n\\, = && \\hspace{-15pt}\n\\Delta_{1,3} \\, A(2,4) - \\Delta_{1,4} \\, A (2,3) \\, , \\quad\n\\nonumber \\\\\n\\bigl[ W(1,2), A^*(3,4) \\bigr]\n\\, = && \\hspace{-15pt}\n\\Delta_{2,3} \\, A^* (1,4) - \\Delta_{2,4} \\, A^* (1,3) \\, , \\quad\n\\nonumber \\\\\n\\bigl[ A^*(1,2), A(3,4) \\bigr]\n\\, = && \\hspace{-15pt}\n\\Delta_{1,3} \\, W(2,4) - \\Delta_{1,4} \\, W (2,3)\n+\\Delta_{2,4} \\, W(1,3)\n\\nonumber \\\\ && \\hspace{-15pt}\n- \\, \\Delta_{2,3} \\, W (1,4) + 2N \\bigl(\\Delta_{12,43}-\\Delta_{12,34}\\bigr)\n\\,. \\quad\n\\end{eqnarray}\nIn particular, $W$\ncoincides with $W_{(2N)}$ in (\\ref{eq2.6a}) and\ngenerates the $u (\\infty,\\infty)$ algebra (of even central charge),\nwhich contains the compact Cartan subalgebra of $so^*\n(4\\infty)$; see Appendix~A. On the other hand, it is\nstraightforward to display the gauge group in the original\npicture as the invariance group of the quaternionic valued bifield\n$Y$~(\\ref{eq2.7}) viewed as a quaternionic form in the\n$N$-dimensional space of real quaternions. We obtain the group\nof $N\\times N$ unitary matrices with quaternionic entries\n\\begin{equation}\\label{eq3.15it}\nU (N,{\\mathbb H}) \\, = \\, Sp(2N) \\,\\equiv\\, USp(2N) \\,,\n\\end{equation}\ni.e., the compact group of unitary complex\nsymplectic $2N \\times 2N$ matrices.\n\n\\section{Unitary positive energy representations and\nsuperselection  structure}\\label{sec4}\n\n\\setcounter{equation}{0}\\setcounter{Theorem}{0}\\setcounter{Definition}{0}\n\nTwo important developments, one in QFT, the other in representation theory,\noriginated half a century ago from the talks of Rudolf Haag and Irving\nSegal at the first Lille conference \\cite{Lille57} on mathematical\nproblems in QFT. Later they gradually drifted apart and lost sight of\neach other. The work of the Hamburg--Rome--G\\\"ottingen school on the\noperator algebra approach to local quantum physics \\cite{H92}\nculminated in the theory of (global) gauge groups and superselection\nsectors \\cite{DR90,BDLR92}. The parallel development of the theory of\nhighest weight modules of semisimple Lie groups (and of the related\ndual pairs) can be traced back from \\cite{EHW83, H89, S90}. Here we aim at\ncompleting the task, undertaken in \\cite{BNRT07} of (restoring and)\ndisplaying the relationship between the two developments.\n\nBefore formulating the main result of this section we shall rewrite\nthe CR (\\ref{eq3.13it}), (\\ref{eq3.14it}) in terms of the discrete\nmodes of $W, A$ and $A^*$ and introduce along the way the conformal\nHamiltonian. We first list the $u(\\infty, \\infty)$ modes of $W$\n\\cite{BNRT07} and write down their CR. Here belong the generators\n$E_{ij}^\\epsilon$ $(\\epsilon=+, -)$ of the maximal compact subalgebra\n$u(\\infty) \\oplus u(\\infty)$ of $u(\\infty, \\infty)$ and of the\nnoncompact raising and lowering operators $X_{ij}$ and $X_{ij}^*$, respectively\n($i,j=1,2,\\dots$)\nsatisfying\n\\begin{eqnarray}\n[E^+_{ij},E^+_{kl}] = \\delta_{jk} E^+_{il} - \\delta_{il} E^+_{kj}, \\hspace{10pt}\n[E^-_{ij},E^-_{kl}] = \\delta_{jk} E^-_{il} - \\delta_{il} E^-_{kj}, \\hspace{10pt}\n[E^+_{ij},E^-_{kl}] = 0, \\hspace{-16pt} \\nonumber\n\\end{eqnarray} \\vskip-9mm\n\\begin{eqnarray}\n[E^+_{ij},X^*_{kl}] = \\delta_{jl} X^*_{ki},\\qquad\n[E^+_{ij},X_{kl}] = - \\delta_{il} X_{kj},\\nonumber \\\\[1mm]\n[E^-_{ij},X^*_{kl}] = \\delta_{jk} X^*_{il}, \\qquad\n[E^-_{ij},X_{kl}] = - \\delta_{ik} X_{jl}, \\nonumber\n\\end{eqnarray} \\vskip-9mm\n\\begin{eqnarray} \\label{eq4.1it}\n[X_{ij},X_{kl}^*] = \\delta_{ik} E^+_{lj} + \\delta_{jl} E^-_{ki} \\;.\n\\end{eqnarray}\nThe commuting diagonal elements $E_{ii}^\\epsilon$ span a compact\n{\\it Cartan subalgebra}. The {\\it antisymmetric bifield} $A$ gives\nrise to an {\\it abelian algebra} spanned by the {\\it raising\n  operators} $Y_{ij}^+ = -Y_{ji}^+$, the {\\it lowering operators}\n$(Y_{ij}^-)^* = -(Y_{ji}^-)^*$ and the operators $F_{ij}$;\nthe modes of $A^*$ are hermitian conjugate to those of $A$.\nThe above $E$'s together with the $F_{ij}$ and their conjugates,\n$F_{ij}^*$, give rise to the maximal compact subalgebra $u(2\\infty)$\nof $so^*(4\\infty)$.\nThe additional nontrivial CR can be restored (applying\nwhen necessary hermitian conjugation) from the following ones:\n\\begin{eqnarray}\\label{eq4.2it}\n[E_{ij}^- , F_{kl}] = && \\hspace{-15pt} \\delta_{jk} F_{il},\\ \\ [F_{ij}, E_{kl}^+] = \\delta_{jk} F_{il},\\ \\\n[F_{ij}, F_{kl}^* ] = \\delta_{jl} E_{ik}^- - \\delta_{ik} E_{lj}^+ ;\n\\nonumber \\\\ {}\n[X_{ij}, F_{kl}] = && \\hspace{-15pt} \\delta_{ik} Y_{jl}^+,\\ \\ [X_{ij}, F_{kl}^* ] = -\\delta_{jl} Y_{ik}^- ,\n\\nonumber \\\\ {}\n[Y_{ij}^\\epsilon , E_{kl}^\\epsilon ] = && \\hspace{-15pt} \\delta_{jk} Y_{il}^\\epsilon - \\delta_{ik}Y_{jl}^\\epsilon,\\ \\\n\\nonumber \\\\ {}\n[Y_{ij}^+ , X_{kl}^* ] = && \\hspace{-15pt} \\delta_{il} F_{kj} - \\delta_{jl} F_{ki},\\ \\\n\\nonumber \\\\ {}\n[Y_{ij}^- , X_{kl}^*] = && \\hspace{-15pt} \\delta_{jk} F_{il}^* - \\delta_{ik} F_{jl}^*;\n\\nonumber \\\\ {}\n[Y_{ij}^\\epsilon , (Y_{kl}^\\epsilon)^* ] = && \\hspace{-15pt} \\delta_{ik} E_{lj}^\\epsilon - \\delta_{jk} E_{li}^\\epsilon\n+ \\delta_{jl} E_{ki}^\\epsilon - \\delta_{il} E_{kj}^\\epsilon;\n\\nonumber \\\\ {}\n[Y_{ij}^+ , F_{kl}^* ] = && \\hspace{-15pt} \\delta_{il} X_{kj} - \\delta_{jl} X_{ki},\\ \\\n\\nonumber \\\\ {}\n[Y_{ij}^- , F_{kl} ] = && \\hspace{-15pt} \\delta_{jk} X_{il} - \\delta_{ik} X_{jl}.\n\\end{eqnarray}\nWe note that the CR (\\ref{eq4.1it}) and (\\ref{eq4.2it}) do not depend\non the ``central charge'' $2N$ of the inhomogeneous terms in\nEqs. (\\ref{eq3.13it}) and (\\ref{eq3.14it}) that is absorbed in the\ndefinition of $E_{ii}^\\epsilon$ (cf.\\ Eq.\\ (\\ref{eqA.3}) of Appendix\nA). The parameter $N$ reappears, however, in the expression for the\n{\\it conformal Hamiltonian} $H_c$ which involves an infinite sum of\nCartan modes -- and hence only belongs to an appropriate completion of\n$u(\\infty, \\infty)\\subset so^*(4\\infty)$:\n\\begin{equation}\n\\label{eq4.3}\nH_c = \\mathop{\\sum}\\limits_{i \\, = \\, 1}^\\infty\\varepsilon_i (E_{ii}^+ + E_{ii}^- - 2N).\n\\end{equation}\nHere the energy eigenvalues $\\varepsilon_i$ form an increasing\nsequence of positive integers (in $D=4$: $\\varepsilon_1 = 1, \\varepsilon_2 =\n\\cdots = \\varepsilon_5 = 2, \\, \\varepsilon_6 = \\cdots =  \\varepsilon_{14} =\n3$, etc.).\nThe {\\it charge} $Q$ and the {\\it number operator} $C_1^u$\nwhich span the centre of $u(\\infty, \\infty)$ and of $u(2\\infty)$,\nrespectively, also involve infinite sums of Cartan modes:\n\\begin{equation}\n\\label{eq4.4}\nQ = \\mathop{\\sum}\\limits_{i \\, = \\, 1}^\\infty (E_{ii}^+ - E_{ii}^-), \\quad\nC_1^u = \\mathop{\\sum}\\limits_{i \\, = \\, 1}^\\infty (E_{ii}^+ + E_{ii}^- - 2N).\n\\end{equation}\nA priori $N$ is a (positive) real number.\nIt has been proven in \\cite{NST02, NRT07},\nhowever, that in a unitary positive energy realization of any algebra of\nbifields generated by local scalar fields of scaling dimension two,\n$N$ must be a natural number.\n\nLet us define the {\\it vacuum representation} of the bifields $W$ and\n$A^{(*)}$ obeying the CR (\\ref{eq3.13it}) and (\\ref{eq3.14it}) as the\n{\\it unitary irreducible positive energy representation}\n(UIPER) of $so^*(4\\infty)$ in which $H_c$ is well\ndefined and has eigenvalue zero on the ground state\n$|vac\\rangle$ (the {\\it vacuum state}). We are now ready to\nstate our main result.\n\n\\medskip\n\n\\noindent {\\bf Theorem 4.1} {\\it In any UIPER {\\rm(}of\\\/ fixed $N${\\rm)} of\\\/\n$so^*(4\\infty)$ we have{\\rm:}\n\n{\\rm(i)} $N$ is a nonnegative integer and all UIPERs of\\\/ $so^*(4\\infty)$ are\nrealized {\\rm(}with multiplicities{\\rm)}\nin the Fock space $\\mathcal{F}_{2N}$ of\\\/\n$2N$ free complex massless scalar fields {\\rm(}see Appendix A{\\rm)}.\n\n{\\rm(ii)} The ground states of equivalent UIPERs of\\\/ $so^*(4\\infty)$ in\n$\\mathcal{F}_{2N}$ form irreducible representations of the gauge group\n$Sp(2N)$. This establishes a one-to-one correspondence between UIPERs\nof\\\/ $so^*(4\\infty)$ occurring in the Fock space and the irreducible\nrepresentations of\\\/ $Sp(2N)$.}\n\n\\medskip\n\nThe {\\it proof} parallels that of Theorem 1 in \\cite[Sect.~2]{BNRT07}\nusing the results of Appendix A. We shall only note that each UIPER of\n$so^*(4\\infty)$ is expressed in terms of the fundamental weights\n$\\Lambda_\\nu$ of $so^*(4n)$ (for large enough $n$, exceeding $N$):\n\\begin{equation}\n\\label{eq4.5}\n\\Lambda = \\mathop{\\sum}\\limits_{\\nu \\, = \\, 0}^{2n-1}k_\\nu\n\\Lambda_\\nu, \\quad k_\\nu\\leq 0.\n\\end{equation}\nIn particular, the vacuum representation has weight $-2N\\Lambda_0$\n(see (\\ref{eqA.16})). Thus, each UIPER remains irreducible when\nrestricted to some $so^*(4n)$, so that we are effectively dealing with\nrepresentations of finite dimensional Lie algebras. We also note that\nthe bifield $W$ has a vanishing vacuum expectation value in view of\n(\\ref{eqA.15}), in accord with its definition as a sum of twist two\nlocal fields.\n\nThe outcome of Theorem 4.1\nand of Theorems 1 and 3 of \\cite{BNRT07} was expected in\nview of the abstract results of the\nDoplicher--Haag--Roberts theory of superselection\nsectors \\cite{H92, DR90, BDLR92}. However,\nconsiderable technical difficulties are\nencountered in relating the extension theory of bifields with the\nrepresentations of the corresponding nets. Our study provides an independent\nderivation of DHR-type results in the field theoretic framework, advancing at\nthe same time the program of classifying globally conformal invariant quantum\nfield theories in four dimensions.\n\n\\section*{Acknowledgments}\n\nThe results of the present paper were reported at three conferences\nduring the summer of 2007: ``LT7 -- Lie Theory and Its Applications in\nPhysics'' (Varna, June 18--23); ``Infinite Dimensional Algebras and\nQuantum Integrable Systems'' (Faro, July 23--27); ``SQS'07 --\nSupersymmetries and Quantum Symmetries'' (Dubna, July 30 -- August\n4). I.T. thanks the organizers of all three events for hospitality and\nsupport. N.M.N.\\ and K.-H.R. also acknowledge the invitation to the\nVarna meeting. B.B.\\ was partially supported by NSF grant\nDMS-0701011. N.M.N.\\ and I.T.\\ were supported in part by the Research\nTraining Network of the European Commission under contract\nMRTN-CT-2004-00514 and by the Bulgarian National Council for Scientific\nResearch under contract PH-1406. K.-H.R.\\ thanks the\nAlexander-von-Humboldt foundation for financial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{se:intro}\n\nThe most widely used extension to rectangular matrices\nof the notion of inverse of square matrices is\nthe so called Moore-Penrose inverse. For a full rank\nmatrix $A\\in\\mathbb{R}^{m\\times n}$ this is defined as\n$A^\\dagger:=(A^{\\rm T} A)^{-1}A^{\\rm T}$ if $m\\geq n$,\nand as $A^\\dagger:=A^{\\rm T} (AA^{\\rm T})^{-1}$, otherwise.\nImmediate applications of $A^\\dagger$ include the solution\nof least square problems\n\\begin{equation}\\label{eq:ls}\n    \\min_{x\\in\\mathbb{R}^n}\\|Ax-b\\|^2,\n\\end{equation}\nwith $b\\in\\mathbb{R}^m$ and $m>n$, or of smallest solutions of\nunderdetermined systems\n\\begin{equation}\\label{eq:us}\n    \\min_{x\\mid Ax=b}\\|x\\|^2\n\\end{equation}\nwhen $n>m$.\nIn both cases, the solution is given by $x=A^\\dagger b$.\nWell known results in error analysis show that the accuracy in the\ncomputation of $A^\\dagger$, or in the computation of the solution\n$x$ for the problems above, crucially depends on the {\\em condition\nnumber} $\\kappa(A):=\\|A\\|\\,\\|A^\\dagger\\|$ of $A$, where $\\|A\\|$\ndenotes the spectral norm (see~\\cite[Ch.~19]{higham:96}). Accuracy\nanalysis is not the only source of interest in $\\kappa(A)$.\nAlgorithms such as the conjugate gradient method produce approximate\nsolutions of linear systems $Px=c$ ---here $P\\in\\mathbb{R}^{m\\times m}$ is a\npositive definite matrix and $c\\in\\mathbb{R}^m$--- with a number of\niterations proportional to $\\sqrt{\\kappa(P)}$ and, in many cases,\nthe matrix $P$ has been obtained as $P=AA^{\\rm T}$ for some matrix\n$A\\in\\mathbb{R}^{m\\times n}$. In those cases, $\\sqrt{\\kappa(P)}=\\kappa(A)$\nand one is again interested in the latter, this time by complexity\nconsiderations.\n\nThe condition number $\\kappa(A)$ is not directly readable\nfrom $A$, and its computation seems to require that\nof $A^\\dagger$. This is a common situation in numerical\nanalysis. A way out of it, proposed as early as 1951\nby von Neumann and Goldstine~\\cite{vNGo51} and\nmore recently pioneered by\nDemmel~\\cite{Demmel88} and Smale~\\cite{Smale97},\nconsists of randomizing the matrix~$A$ ---say, by endowing\n$\\mathbb{R}^{m\\times n}$ with a multivariate standard Gaussian\ndistribution $N(0,\\mathrm{I})$--- and considering\nits condition number as a derived random variable.\n\nIn Chen and Dongarra~\\cite{ChDo:05} the\nfollowing tail estimates on $\\kappa(A)$ were shown\nfor $A\\in\\mathbb{R}^{m\\times n}$ with $n\\ge m$:\nfor $x\\ge n-m+1$ we have\n\\begin{equation}\\label{eq:chen-dong}\n \\frac1{\\sqrt{2\\pi}}\\, \\Big(\\frac{1}{5x}\\Big)^{n-m+1}\\ \\leq\\\n \\mathop{\\rm Prob}_{A\\sim N(0,\\mathrm{I})}\n \\Big\\{\\kappa(A)\\geq\\frac{x}{1-\\lambda}\\Big\\}\\\n  \\leq\\ \\frac1{\\sqrt{2\\pi}}\\, \\Big(\\frac{7}{x}\\Big)^{n-m+1} .\n\\end{equation}\nMoreover, the expectation $\\mathbb{E}(\\kappa(A))$ can be bounded as a function\nof the {\\em elongation} $\\frac{m-1}{n}$ only, independently of~$n$.\n(We remark that this is not true for Demmel's scaled\ncondition number $\\|A\\|_F\\,\\|A^\\dagger\\|$, compare~\\cite{edelm:92}.)\nMore precisely,\nfor a sequence $(m_n)$ of integers such that\n$\\lim_{n\\to\\infty}m_n\/n=\\lambda\\in(0,1)$ and a\nsequence of standard Gaussian random matrices $A_n \\in\\mathbb{R}^{m_n\\times n}$,\nwe have in almost sure convergence\n\\begin{equation}\\label{eq:Edelman_kappa}\n  \\kappa(A_n)\\stackrel{\\rm a.s.}{\\longrightarrow}\n  \\frac{1+\\sqrt{\\lambda}}{1-\\sqrt{\\lambda}} .\n\\end{equation}\nThis follows from Geman~\\cite{geman:80} and Silverstein~\\cite{silver:85}\n(see Edelman~\\cite{edelm:88} for more precise results).\n\nThe above results provide theoretical reasons of why least squares\nproblems such\nas~\\eqref{eq:ls} or underdetermined systems such as~\\eqref{eq:us}\nare solved to great accuracy or why the conjugate gradient method is\nso efficient in practice. In fact, it follows from~\\eqref{eq:Edelman_kappa}\nthat the expected number of iterations of\nthe conjugate gradient method on the random input $P=AA^{\\rm T} $ remains\nbounded in terms of the {\\em elongation} $m\/n$ as $n\\to\\infty$ and\n$A\\in\\mathbb{R}^{m\\times n}$ is standard Gaussian. Our main result stated\nbelow implies that this phenomenon is still true for any matrix\nthat is only slightly perturbed.\n\nThe choice of $N(0,\\mathrm{I})$ as underlying data distribution\nis pervasive in the {\\em average-case analysis} of condition\nnumbers\n(and other quantities occurring in numerical analysis). It has the\nvirtue of simplicity as a first approach to understanding\nwhich condition numbers one may expect. But it has been\ncriticized due to the loose relationship of the Gaussian\n$N(0,\\mathrm{I})$ to the measures that may be governing\ndata drawing in practice.\nIn particular, it has been observed\nthat the use of Gaussians may be `optimistic' in the sense\nthat they may put more probability mass on the instances where\nthe values of the function $\\psi$ under consideration are small.\nSuch an optimism would produce yield an expectation\n$\\mathbb{E}(\\psi)$ smaller than the true one.\n\nAn alternate, more conservative,\nform of analysis has been proposed by Spielman and Teng\nunder the name of {\\em smoothed analysis}. It replaces\nthe Gaussian measure $N(0,\\mathrm{I})$ by the measures\n$N(\\overline{A},\\sigma^2\\mathrm{I})$ where $\\overline{A}$ is arbitrary. The idea is\nthen to replace the unlikely `average data' by a (usually\nsmall) perturbation of any possible occurring data. The\nrationale for this form of analysis is offered in a number\nof papers~\\cite{ST-simplex:04,sst:06,ST:06,ST:09} and\nwe won't repeat it here in full.\nWe note, nonetheless,\nthat the local nature of randomization in smoothed analysis,\ncoupled with its worst-case dependence on the input data,\nremoves from smoothed analysis the possible optimism\nwe mentioned above for average-case analysis.\nIn recent years,\ndifferent aspects of algorithm behavior for a variety of\nproblems have been analyzed this way. These include\ncondition numbers of square matrices with\nreal~\\cite{Wsch:04} or $\\{-1,1\\}$\ncoefficients~\\cite{TaoVu:07}, complexity of\ninterior-point methods~\\cite{DuSpTe:09}, and machine\nlearning~\\cite{ArVa:09}. The typical satisfying result\nis {\\em polynomial smoothed complexity}\n(see~\\cite[Def.~2]{ST:09}), consisting of a bound\nof the form\n\\begin{equation}\\label{eq:sc}\n  \\sup_{\\overline{A}} \\mathbb{E}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})} \\psi(A)\n  \\leq c \\sigma^{-k_1} \\mathsf{size}(A)^{k_2}\n\\end{equation}\nwhere $\\psi$ is the function whose behavior we are\nanalyzing and $c,k_1,k_2$ are positive constants.\n\nIn this paper we provide a smoothed analysis for\nMoore-Penrose inversion, extending~\\eqref{eq:chen-dong}\nfrom the average-case analysis to smoothed analysis.\nTo state the results we need to introduce some notations.\nWe assume $1\\le m\\le n$ throughout the paper.\nFor a standard Gaussian $X\\in\\mathbb{R}^{m\\times n}$ we put\n\\begin{equation}\\label{eq:defQ}\n Q(m,n) := \\frac1{\\sqrt{n}}\\, \\mathbb{E}(\\|X\\|) .\n\\end{equation}\n(Lemma~\\ref{pro:Espnorm} shows that $Q(m,n) \\le 6$ .)\nWe define for $\\lambda \\in (0,1)$ the quantity\n\\begin{equation}\\label{eq:defc}\n c(\\lambda) \\ :=\\   \\sqrt{\\frac{1+\\lambda}{2(1-\\lambda)}} .\n\\end{equation}\nNote that $c(\\lambda)$ is monotonically increasing,\n$\\lim_{\\lambda\\to 0}c(\\lambda) = \\frac1{\\sqrt{2}}$ and\n$\\lim_{\\lambda\\to 1}c(\\lambda) = \\infty$.\nFurther, for $1\\le m\\le n$ and $0<\\sigma\\le 1$, we\ndefine the {\\em elongation}\n$\\lambda := \\frac{m-1}{n}$ and introduce the quantity\n\\begin{equation}\\label{eq:defB}\n \\zeta_\\sigma(m,n) \\ :=\\ \\Big(Q(m,n) + \\frac1{\\sigma\\sqrt{n}}\\Big)\\\n  c(\\lambda)^{\\frac1{n-m+1}}  .\n\\end{equation}\n\nOur main result is the following tail bound on the condition\nnumber of rectangular matrices under local Gaussian perturbations.\n\n\\begin{theorem}\\label{thm:tailbound}\nSuppose that $\\overline{A}\\in\\mathbb{R}^{m\\times n}$ satisfies\n$\\|\\overline{A}\\| \\le 1$ and let $0<\\sigma\\le 1$.\nPut $\\lambda := \\frac{m-1}{n}$.\nThen, for $z \\ge \\zeta_\\sigma(m,n)$, we have \n$$\n  \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\n   \\Big\\{\\kappa(A)\\geq\\frac{ez}{1-\\lambda}\\Big\\}\\\n  \\leq\\\n   2c(\\lambda)\\bigg[\n   \\Big(Q(m,n) + \\sqrt{2\\ln(2z)} + \\frac1{\\sigma\\sqrt{n}}\\Big)\\,\n    \\frac1z\\bigg]^{n-m+1} .\n$$\n\\end{theorem}\n\n\\begin{remark}\n{\\bf 1.} The decay in~$z$ in this tail bound is the same as\nin~\\eqref{eq:chen-dong} up to the logarithmic factor $\\sqrt{\\ln z}$.\nWe believe that the latter is an artefact of our proof that could be omitted.\nIn fact, the exponent $n-m+1$ is just the codimension of the set\n$\\Sigma := \\{A\\in\\mathbb{R}^{m\\times n} \\mid \\mathrm{rk} A < m\\}$\nof rank deficient matrices, cf.~\\cite{harr:95}.\nMoreover, it is known~\\cite{higham:96} that\n$\\|A^\\dagger\\| = 1\/\\mathrm{dist}(A,\\Sigma)$\nwhere the distance is measured in the Euclidean norm.\nFrom the interpretation of $\\mathop{\\rm Prob}\\{\\kappa(A)\\geq t\\}$\nas the volume of a tube around $\\Sigma$, as discussed in~\\cite{BCL:06a},\none would therefore expect a decay of order $1\/z^{n-m+1}$.\n\n{\\bf 2.} When $\\sigma=1$ and $\\overline{A}=0$, Theorem~\\ref{thm:tailbound}\nyields tail bounds for the usual average case. One may therefore\ncompare these bounds with~\\eqref{eq:chen-dong}.\nIn doing so, we see that\nthe bound in Theorem~\\ref{thm:tailbound}\nhas the additional factor~$c(\\lambda)$ (going to $\\infty$\nfor $\\lambda\\to 1$). However, we note that the bound~\\eqref{eq:chen-dong}\nholds only for\n$x=ez\\ge n-m + 1$,\nwhile our bound holds for any $z \\ge \\zeta_\\sigma(m,n)$.\nFurthermore, if we fix\n$\\lambda\\in(0,1)$ and let $(m_n)$ be a sequence of positive integers\nsuch that $\\lim m_n\/n=\\lambda$,\nit follows from~\\cite{geman:80} that\n$$\n\\lim_{n\\to\\infty} Q(m_n,n)= 1 + \\sqrt{\\lambda}.\n$$\nThis implies that\n$\\lim_{n\\to\\infty} \\zeta_\\sigma(m_n,n) = 1 + \\sqrt{\\lambda}$\nfor fixed $\\sigma \\in (0,1]$ and, in particular, that\n$\\zeta_\\sigma(m_n,n) \\le 2$ for sufficiently large~$n$ .\nThat is, for large $n$, the tail bound in Theorem~\\ref{thm:tailbound}\nis valid for any $z\\ge 2$.\n\\end{remark}\n\nTheorem~\\ref{thm:tailbound} easily implies the following\nbound on expectations.\n\n\\begin{corollary}\\label{cor:main}\nFor all $\\lambda_0\\in(0,1)$\nthere exists $n_0$ such that for all\n$1\\le m\\le n$ such that\n$\\lambda = \\frac{m-1}{n} \\le \\lambda_0$ and\n$n\\ge n_0$ we have\nfor all $\\sigma$ with $\\frac1{\\sqrt{m}}\\le \\sigma\\le 1$,\nand all\n$\\overline{A}\\in\\mathbb{R}^{m\\times n}$ with $\\|\\overline{A}\\| \\le 1$, that\n$$\n \\mathbb{E}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}(\\kappa(A))\n  \\ \\leq\\ \\frac{20.1}{1-\\lambda} .\n$$\n\\end{corollary}\n\nAs for the average-case analysis, this bound\nis independent of~$n$ and depends only on the bound $\\lambda_0$\non the elongation.\nThus we have a bound of type~\\eqref{eq:sc} with $k_2=0$.\nSurprisingly, the smoothed complexity bound in Corollary~\\ref{cor:main}\nis also independent of~$\\sigma$. We thus add reasons\n---and we will become more specific in\nSection~\\ref{sec:applications}---\nto the current understanding of the accuracy\nin least squares or underdetermined system solving\nor the complexity of the conjugate gradient method.\n\nA first approach to the smoothed analysis of Moore-Penrose\ninversion appears in~\\cite{CDW:05}. The bounds obtained in\nthat paper are worse by an order of magnitude\nthan those we obtain here.\nIn Section~\\ref{sec:numerical} we compare these bounds\n{\\sf with ours} \nas well as with actual averages obtained,\nfor specific values of $n,m$ and $\\sigma$, in numerical\nsimulations.\n\nOur proof techniques are an extension of methods\nemployed by Sankar et al.~\\cite{sst:06}.\n\n\\medskip\n\n\\noindent{\\bf Acknowledgements.}\nThis work was carried out during the special semester on Foundations\nof Computational Mathematics in the fall of 2009.\nWe thank the Fields Institute in Toronto for hospitality and\nfinancial support.\n\n\\section{Preliminaries}\n\n\\subsection{Some definitions and notation}\n\nThe {\\em spectral norm} of a matrix $A\\in\\mathbb{R}^{m\\times n}$ is defined as\n$\\|A\\| := \\sup_{\\|x\\|=1} \\|Ax\\|$, where $\\|x\\|$ denotes the Euclidean norm.\nThe {\\em Frobenius norm} of $A$ is defined as the Euclidean norm of $A$\nwhen interpreted as a vector.\n\nSuppose that $A\\in\\mathbb{R}^{m\\times n}$ is of maximal rank and $m\\le n$.\nThe {\\em Moore-Penrose inverse} of $A$\nis defined as\n$A^\\dagger := A^{\\rm T}  (AA^{\\rm T})^{-1}\\in\\mathbb{R}^{n\\times m}$.\nIt can also be characterized as follows.\nFor any $v\\in\\mathbb{R}^m$ the vector $w=A^\\dagger v$ is orthogonal to\nthe kernel of~$A$ and satisfies $Aw=v$.\nThe {\\em condition number} $\\kappa(A)$ is defined as\n$\\kappa(A) :=\\|A\\| \\cdot \\|A^\\dagger\\|$.\n\nLet $\\overline{A}\\in \\mathbb{R}^{m\\times n}$ and $\\sigma>0$.\nThe {\\em isotropic normal distribution} $N(\\overline{A},\\sigma \\mathrm{I})$ with center~$\\overline{A}$\nand covariance matrix $\\sigma^2\\mathrm{I}$ is the probability distribution on\n$\\mathbb{R}^{m\\times n}$ with the density\n$$\n \\rho_{\\overline{A},\\sigma}(A) := \\frac1{(2\\pi)^{\\frac{mn}{2}}}\\,\n  e^{-\\frac{\\|A-\\overline{A}\\|_F^2}{2\\sigma^2}} .\n$$\n\n\\begin{lemma}\\label{le:gbound}\nFor $\\lambda\\in (0,1)$ we have\n$\\lambda^{-\\frac{\\lambda}{1-\\lambda}} \\le e$.\n\\end{lemma}\n\n{\\noindent\\sc Proof. \\quad}\nWriting $u=1\/\\lambda$ the assertion is equivalent to\n$u^{\\frac1{u-1}} \\le e$ or $u\\le e^{u-1}$,\nwhich is certainly true for $u \\ge 1$.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\subsection{Concentration on spheres}\n\nLet $\\mathbb{S}^{m-1}:=\\{x\\in\\mathbb{R}^m \\mid \\|x\\|=1\\}$ denote the unit sphere in $\\mathbb{R}^m$.\nWe denote by ${\\cal O}_{m-1}$ its volume, which is given by\n${\\cal O}_{m-1} = 2\\pi^{m\/2}\/\\Gamma(\\frac{m}{2})$.\n\nThe following estimate tells us how likely a random point on $\\mathbb{S}^{m-1}$\nwill lie in a fixed spherical cap.\n\n\\begin{lemma}\\label{lem:s}\nLet $u\\in\\mathbb{S}^{m-1}$ be fixed, $m\\geq 2$. Then, for all $\\xi\\in[0,1]$,\n$$\n   \\mathop{\\rm Prob}_{v\\sim U(\\mathbb{S}^{m-1})}\n   \\big\\{\\big|u^{\\rm T} v\\big|\\geq \\xi\\big\\} \\geq\n   \\sqrt{\\frac{2}{\\pi m}}\\; (1-\\xi^2)^{\\frac{m-1}{2}}.\n$$\n\\end{lemma}\n\n{\\noindent\\sc Proof. \\quad}\nWe put $\\theta=\\arccos\\xi$ and let $\\mathsf{cap}(u,\\theta)$ denote\nthe spherical cap in $\\mathbb{S}^{m-1}$ with center~$u$ and angular radius~$\\theta$.\nUsing the bounds in Lemmas~2.1 and 2.2 of~\\cite{bcl:08a} we get\n$$\n   \\mathop{\\rm Prob}_{v\\sim U(\\mathbb{S}^{m-1})}\n   \\big\\{\\big|u^{\\rm T} v\\big|\\geq \\xi\\big\\}\n  = \\frac{2\\,\\mathsf{vol\\,}\\mathsf{cap}(u,\\theta)}{\\mathsf{vol\\,}\\mathbb{S}^{m-1}}\n  \\geq \\frac{2{\\cal O}_{m-2}}{{\\cal O}_{m-1}}\\;\n   \\frac{(1-\\xi^2)^{\\frac{m-1}{2}}}{(m-1)}.\n$$\nUsing the formula for ${\\cal O}_{m-1}$ and the recursion\n$\\Gamma(x+1)=x\\Gamma(x)$ we have\n$$\n    \\frac{{\\cal O}_{m-2}}{{\\cal O}_{m-1}}\n  = \\frac{1}{\\sqrt{\\pi}}\n   \\frac{\\Gamma\\big(\\frac{m}{2}\\big)}\n   {\\Gamma\\big(\\frac{m-1}{2}\\big)}\n  = \\frac{1}{\\sqrt{\\pi}}\n   \\frac{\\Gamma\\big(\\frac{m+1}{2}\\big)}\n   {\\Gamma\\big(\\frac{m-1}{2}\\big)}\n   \\frac{\\Gamma\\big(\\frac{m}{2}\\big)}\n   {\\Gamma\\big(\\frac{m+1}{2}\\big)}\n   = \\frac{m-1}{2\\sqrt{\\pi}}\n   \\frac{\\Gamma\\big(\\frac{m}{2}\\big)}{\\Gamma\\big(\\frac{m+1}{2}\\big)}.\n$$\nThe assertion follows now from the estimate\n\\begin{equation}\\label{eq:Gine}\n \\frac{\\Gamma\\big(\\frac{m}{2}\\big)}{\\Gamma\\big(\\frac{m+1}{2}\\big)}\n \\ \\ge\\ \\sqrt{\\frac{2}{m}}.\n\\end{equation}\nThis estimate can be quickly seen as follows.\nSuppose that $Z\\in\\mathbb{R}^{m}$ is standard normal distributed.\nUsing polar coordinates and the variable\ntransformation $u=\\rho^2\/2$ we get\n\\begin{eqnarray}\\notag\n \\mathbb{E}(\\|Z\\|) &= &\\frac{{\\cal O}_{m-1}}{(2\\pi)^{\\frac{m}{2}}}\n \\int_0^\\infty \\rho^m e^{-\\frac{\\rho^2}{2}} d\\rho\n \\:=\\: \\frac{{\\cal O}_{m-1}}{(2\\pi)^{\\frac{m}{2}}}\\, 2^{\\frac{m-1}{2}}\n    \\int_0^\\infty u^{\\frac{m-1}{2}} e^{-u} du \\\\ \\label{eq:remember}\n &=& \\frac{{\\cal O}_{m-1}}{(2\\pi)^{\\frac{m}{2}}}\\, 2^{\\frac{m-1}{2}}\n    \\Gamma(\\frac{m+1}{2})\n \\:=\\: \\sqrt{2}\\,\\frac{\\Gamma(\\frac{m+1}{2})}\n   {\\Gamma(\\frac{m}{2})},\n\\end{eqnarray}\nwhere we used the definition of the Gamma function\nfor the second last equality.\nTo complete the proof of \\eqref{eq:Gine} we note that\n$\\mathbb{E}(\\|Z\\|) \\le \\sqrt{\\mathbb{E}(\\|Z\\|^2)} =\\sqrt{m}$.\n{\\mbox{}\\hfill\\qed}\\medskip\n\nFor later use we note that~\\eqref{eq:remember} implies\n$$\n \\frac{\\Gamma\\big(\\frac{m+1}{2}\\big)}{\\Gamma\\big(\\frac{m}{2}\\big)}\n = \\frac{\\Gamma\\big(\\frac{m+2}{2}\\big)}{\\Gamma\\big(\\frac{m}{2}\\big)}\n      \\frac{\\Gamma\\big(\\frac{m+1}{2}\\big)}{\\Gamma\\big(\\frac{m+2}{2}\\big)}\n = \\frac{m}{2}\\, \\frac{\\Gamma\\big(\\frac{m+1}{2}\\big)}{\\Gamma\\big(\\frac{m+2}{2}\\big)}\n \\ \\ge\\  \\frac{m}{2}\\, \\sqrt{\\frac{2}{m+1}},\n$$\nusing~\\eqref{eq:Gine} for the right-hand inequality. Therefore\n\\begin{equation}\\label{eq:chi-lb}\n \\mathbb{E}(\\|Z\\|)\\ \\ge\\ \\frac{m}{\\sqrt{m+1}} .\n\\end{equation}\n\n\\subsection{Large deviations}\n\nWe will use a powerful large deviation result.\nLet $F:\\mathbb{R}^N\\to\\mathbb{R}$ be a Lipschitz continous function with Lipschitz\nconstant~$L$, so that $|F(x)-F(y)| \\le L \\|x-y\\|$ for all $x,y\\in\\mathbb{R}^N$,\nwhere $\\|\\ \\|$ denotes the Euclidean norm.\nNow suppose that $x\\in\\mathbb{R}^N$ is a standard Gaussian random vector such that\n$\\mathbb{E}(F(x))$ exists. Then it is known~\\cite[(1.4)]{letala:91} that for all $t>0$\n\\begin{equation}\\label{eq:concentration}\n  \\mathop{\\rm Prob}\\{F(x)\\geq \\mathbb{E}(F) +t\\}\\leq e^{-\\frac{t^2}{2L^2}}.\n\\end{equation}\n(We note that in ~\\cite[(1.4)]{letala:91} this is only stated for the median, but\nthe inequality holds as well for the expectation. See also~\\cite{ledoux:01}.)\n\n\\subsection{A bound on the expected spectral norm}\n\nThe function $\\mathbb{R}^{m\\times n}\\to\\mathbb{R}$ mapping a matrix $X$ to its\nspectral norm~$\\|X\\|$ is Lipschitz continuous with Lipschitz constant~$1$,\nas $\\|X-Y\\| \\le \\|X-Y\\|_F$.\nThe concentration bound~\\eqref{eq:concentration}, together\nwith~\\eqref{eq:defQ}, implies that for $t>0$,\n\\begin{equation}\\label{eq:spnb}\n\\mathop{\\rm Prob}\\Big\\{\\|X\\| \\ge Q(m,n)\\sqrt{n} + t\\}\\ \\le\\ e^{-\\frac{t^2}{2}} .\n\\end{equation}\nThis tail bound easily implies the following large deviation result.\n\n\\begin{proposition}\\label{le:enorm}\nLet $\\overline{A}\\in\\mathbb{R}^{m\\times n}$ with $m\\le n$, $\\|\\overline{A}\\| \\le 1$, and $\\sigma\\in(0,1]$.\nIf $A\\in\\mathbb{R}^{m\\times n}$ follows the law $N(\\overline{A},\\sigma^2\\mathrm{I})$, then, for $t>0$,\n$$\n \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\Big\\{\\|A\\|\\geq Q(m,n)\\sigma\\sqrt{n}+t+1\\Big\\}\n \\ \\leq\\  e^{-\\frac{t^2}{2\\sigma^2}}.\n$$\n\\end{proposition}\n\n{\\noindent\\sc Proof. \\quad}\nWe note that\n$\\|A\\|\\geq Q(m,n)\\sigma\\sqrt{n}+t+1$ implies that\n$\\|A -\\overline{A}\\| \\geq \\|A\\| -\\|\\overline{A}\\| \\ge Q(m,n)\\sqrt{n} + t$.\nMoreover, if $A\\in\\mathbb{R}^{m\\times n}$ follows the law $N(\\overline{A},\\sigma^2\\mathrm{I})$, then\n$X:=\\frac{A-\\overline{A}}{\\sigma}$ is standard Gaussian in $\\mathbb{R}^{m\\times n}$.\nThe assertion follows from~\\eqref{eq:spnb}.\n{\\mbox{}\\hfill\\qed}\\medskip\n\nWe derive now an upper bound on $Q(m,n)$.\nSuch result should be well-known but we could not locate in the literature.\n\n\\begin{lemma}\\label{pro:Espnorm}\nFor $n>1$ we have\n$\\sqrt{\\frac{n}{n+1}}\n\\le Q(m,n)\\le\n 2\\Big( 1 + \\sqrt{\\frac{2\\ln (2m-1)}{n}} + \\frac1{\\sqrt{n}}\\Big) \\le 6$.\n\\end{lemma}\n\nThe proof relies on the following lemma.\n\n\\begin{lemma}\\label{le:maxchi}\nLet $r_1,\\ldots,r_n$ be independent random variables with nonnegative values\nsuch that $r_i^2$ is $\\chi^2$-distributed with $f_i$ degrees of freedom.\nThen,\n$$\n \\mathbb{E}\\Big(\\max_{1\\le i \\le n} r_i\\Big) \\le \\max_{1\\le i\\le n}\\sqrt{f_i}\n   + \\sqrt{2\\ln n} +  1 .\n$$\n\\end{lemma}\n\n{\\noindent\\sc Proof. \\quad}\nWe start by a large deviation estimate for $\\chi^2$-distributed random variables.\nNote that $\\mathbb{R}^f\\to\\mathbb{R}$, $x\\mapsto \\|x\\|$, is Lipschitz continuous with\nLip\\-schitz constant~$1$.\nFrom \\eqref{eq:concentration} we know that for standard Gaussian $x\\in\\mathbb{R}^n$\nand all $t>0$,\n$$\n  \\mathop{\\rm Prob}\\{\\|x\\| \\geq \\mathbb{E}(\\|x\\|) +t\\}\\ \\leq\\ e^{-\\frac{t^2}{2}}.\n$$\nSince $\\mathbb{E}(\\|x\\|)\\le \\sqrt{\\mathbb{E}(\\|x\\|^2)} = \\sqrt{f}$, this implies for all $t>0$,\n\\begin{equation}\\label{eq:chi-conc}\n  \\mathop{\\rm Prob}\\{\\|x\\| \\geq \\sqrt{f} +t \\}\\ \\leq\\ e^{-\\frac{t^2}{2}}.\n\\end{equation}\n\nWe suppose now that $r_1,\\ldots,r_n$ are\nindependent random variables with nonnegative values\nsuch that $r_i^2$ is $\\chi^2$-distributed with $f_i$ degrees of freedom.\nPut $f:=\\max_i f_i$.\nEquation~\\eqref{eq:chi-conc} tells us that for all $i$ and all $t>0$,\n$$\n \\mathop{\\rm Prob}\\{ r_i \\ge \\sqrt{f} + t \\}\\ \\le\\  e^{-\\frac{t^2}{2}}\n$$\nand hence, by the union bound,\n$$\n \\mathop{\\rm Prob}\\Big\\{\\max_{1\\le i\\le n} r_i \\ge \\sqrt{f} + t \\Big\\}\\\n  \\le\\ n e^{-\\frac{t^2}{2}} .\n$$\nFor a fixed parameter~$b\\ge 1$ (to be determined later), this implies\n\\begin{eqnarray*}\n \\mathbb{E}(\\max_{1\\le i\\le n} r_i) &\\le& \\sqrt{f} + b + \\int_{\\sqrt{f} + b}^\\infty\n                       \\mathop{\\rm Prob}\\{\\max_{1\\le i\\le n} r_i \\ge T \\}\\, dT \\\\\n  &=& \\sqrt{f} + b + \\int_{b}^\\infty \\mathop{\\rm Prob}\\{\\max_{1\\le i\\le n}  r_i \\ge \\sqrt{f} + t \\}\\, dt \\\\\n  &\\le& \\sqrt{f} + b + n \\int_{b}^\\infty  e^{-\\frac{t^2}{2}}\\,dt .\n\\end{eqnarray*}\nUsing the well-known estimate\n$$\n \\frac1{\\sqrt{2\\pi}}\\int_b^\\infty  e^{-\\frac{t^2}{2}}\\,dt\n \\le \\frac1{b\\sqrt{2\\pi}}\\,e^{-\\frac{b^2}{2}}\n  \\le \\frac1{\\sqrt{2\\pi}}\\,e^{-\\frac{b^2}{2}}\n$$\nwe obtain\n$$\n \\mathbb{E}(\\max_{1\\le i\\le n} r_i) \\le \\sqrt{f} + b + n e^{-\\frac{b^2}{2}} .\n$$\nFinally, choosing $b:=\\sqrt{2\\ln n}$ we get\n$$\n \\mathbb{E}(\\max_{1\\le i\\le n} r_i) \\le \\sqrt{f} + \\sqrt{2\\ln n} + 1,\n$$\nas claimed.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\proofof{Lemma~\\ref{pro:Espnorm}}\nA general matrix $X\\in\\mathbb{R}^{m\\times n}$ can be transformed into a\nbidiagonal matrix of the form\n$$\nY:=\\begin{bmatrix}\nv_n    &       &         &      & 0     &\\cdots &0     \\\\\nw_{m-1}&v_{n-1}&         &      & \\vdots&       &\\vdots\\\\\n       &\\ddots &\\ddots   &      & \\vdots&       &\\vdots\\\\\n       &       & w_1     &v_{n-m+1}&0   &\\cdots &0\n\\end{bmatrix}\n$$\nwith $v_i,w_j\\ge 0$ by performing Householder transformations\nfrom the left and right hand side of~$X$, cf.~\\cite[\\S5.4.3]{golloan:83}.\nIn particular, $\\|X\\| = \\|Y\\|$.\nAn analysis of this transformation shows that if we start\nwith a standard Gaussian matrix $X$, then\nthe $v_n,\\ldots,v_{n-m+1},w_{m-1},\\ldots,w_1$\nare independent random variables such that $v_i^2$ and $w_i^2$ are\n$\\chi^2$-distributed with $i$~degrees of freedom, cf.~\\cite{silver:85}.\n\nThe spectral norm of $Y$ is bounded by\n$\\max_{i}v_i + \\max_j w_j \\le 2 r$, where\n$r$~denotes the maximum of the values $v_i$ and $w_j$.\nLemma~\\ref{le:maxchi} implies that, for $n>1$,\n$$\n \\mathbb{E}(r) \\le \\sqrt{n} + \\sqrt{2\\ln (2m-1)} + 1\n  \\le 3\\sqrt{n} .\n$$\nThis shows the claimed upper bound on $Q(m,n)$.\nFor the lower bound we note that\n$\\|Y\\| \\ge |v_n|$ which gives\n$\\mathbb{E}(\\|Y\\|) \\ge \\mathbb{E}(|v_n|)$.\nThe claimed lower bound now follows from~\\eqref{eq:chi-lb},\nwhich states that\n$\\mathbb{E}(|v_n|) \\ge \\sqrt{\\frac{n}{n+1}}$.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\section{Proof of the main results}\n\nThe main work consists of deriving tail bounds on $\\|A^\\dagger\\|$,\nwhich is done in the next subsection.\n\n\\subsection{Tail bounds for $\\|A^\\dagger\\|$}\\label{se:tb-Adag}\n\n\n\\begin{proposition}\\label{thm:main_tail}\nLet $\\overline{A}\\in\\mathbb{R}^{m\\times n}$, $\\sigma>0$, and put\n$\\lambda :=\\frac{m-1}{n}$.\nFor random $A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})$\nwe have, for any $t>0$,\n$$\n \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\Big\\{\\|A^\\dagger\\|\\geq \\frac{t}{1-\\lambda}\\Big\\}\n  \\ \\leq\\ c(\\lambda)\\,\n       \\bigg(\\frac{e}{\\sigma\\sqrt{n}\\, t}\\bigg)^{(1-\\lambda)n}.\n$$\n\\end{proposition}\n\nWe first show the following result.\n\n\\begin{proposition}\\label{prop:bound1}\nFor all $v\\in \\mathbb{S}^{m-1}$,\n$\\overline{A}\\in\\mathbb{R}^{m\\times n}$, $\\sigma>0$, and $\\xi>0$\nwe have\n$$\n  \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\n  \\big\\{\\|A^\\dagger v\\|\\geq \\xi \\big\\}\\ \\leq\\\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m+1}}\\,\n               \\frac{{\\cal O}_{n-m}}{n-m+1}\\,\n               \\Big(\\frac{1}{\\sigma \\xi}\\Big)^{n-m+1}.\n$$\n\\end{proposition}\n\n\n{\\noindent\\sc Proof. \\quad}\nWe first claim that, because of unitary invariance,\nwe may assume that $v=e_m:=(0,\\ldots,0,1)$.\nTo see this, take $\\Phi\\in U(m)$ such that $v=\\Phi e_m$.\nConsider the isometric map\n$A\\mapsto B=\\Phi^{-1}A$ which transforms the density\n$\\rho_{\\overline{A},\\sigma}(A)$ into a density of the same form, namely\n$\\rho_{\\Phi^{-1}\\overline{A},\\sigma}(B)$.\nThus the assertion for $e_m$ and random $B$\nimplies the assertion for $v$ and $A$, noting that\n$A^\\dagger v=B^\\dagger e_m$. This proves the claim.\n\nWe are going to characterize the norm of $w:=A^\\dagger e_m$\nin a geometric way.\nLet $a_i$ denote the $i$th row of $A$. Almost surely,\nthe rows $a_1,\\ldots,a_m$ are linearly independent;\nhence, we assume so in what follows.\nLet\n$$\n  R:=\\mathsf{span}\\{a_1,\\ldots,a_{m}\\},\\ S:=\\mathsf{span}\\{a_1,\\ldots,a_{m-1}\\} .\n$$\nLet $S^\\perp$ denote the orthogonal\ncomplement of $S$ in $\\mathbb{R}^n$. We decompose\n$a_m=a_m^\\perp + a_m^S$,\nwhere $a_m^\\perp$ denotes the orthogonal projection\nof $a_m$ onto $S^\\perp$ and $a_m^S\\in S$. Then $a_m^\\perp\\in R$\nsince both $a_m$ and $a_m^S$ are in $R$. It follows that\n$a_m^\\perp\\in R\\cap S^\\perp$.\n\nWe claim that $w\\in R\\cap S^\\perp$ as well. Indeed,\nnote that $R$ equals the orthogonal complement of the\nkernel of $A$ in $\\mathbb{R}^n$. Therefore, by definition of the\nMoore-Penrose inverse, $w=A^\\dagger e_m$ lies in $R$.\nMoreover, since $A A^\\dagger=\\mathrm{I}$, we have\n$\\langle w,a_i\\rangle =0$ for $i=1,\\ldots,m-1$\nand hence $w\\in S^\\perp$ as well.\n\nIt is immediate to see that $\\dim R\\cap S^\\perp=1$.\nIt then follows that $R\\cap S^\\perp=\\mathbb{R} w = \\mathbb{R} a_m^\\perp$.\nSince $\\langle w,a_m\\rangle =1$, we get\n$1 = \\langle w,a_m\\rangle = \\langle w,a_m^\\perp \\rangle\n      = \\|w\\|\\, \\|a_m^\\perp\\|$\nand therefore\n\\begin{equation}\\label{eq:star}\n \\|A^{\\dagger}e_m\\| = \\frac1{\\|a_m^\\perp\\|}.\n\\end{equation}\n\nLet $A_m\\in\\mathbb{R}^{(m-1)\\times n}$ denote the matrix\nobtained from~$A$ by omitting $a_m$. The density\n$\\rho_{\\overline{A},\\sigma}$ factors as\n$\\rho_{\\overline{A},\\sigma}(A) =\\rho_1(A_n)\\rho_2(a_n)$ where\n$\\rho_1$ and $\\rho_2$ denote the density functions of\n$N(\\overline{A}_m,\\sigma^2\\mathrm{I})$ and\n$N(\\bar{a}_m,\\sigma^2\\mathrm{I})$, respectively\n(the meaning of $\\overline{A}_m$ and $\\bar{a}_m$\nbeing clear). Fubini's Theorem combined\nwith \\eqref{eq:star} yield, for $\\xi>0$,\n\\begin{eqnarray}\\label{eq:WX}\n  \\mathop{\\rm Prob}_{N(\\overline{A},\\sigma^2\\mathrm{I})}\\big\\{\\|A^{\\dagger}e_m\\|\\geq \\xi\\big\\}\n   &=&\n     \\int_{\\|A^{\\dagger}e_m\\| \\ge \\xi}\n     \\rho_{\\overline{A},\\sigma^2\\mathrm{I}}(A)\\, dA\\\\\n  &=&\n     \\int_{A_{m}\\in\\mathbb{R}^{(m-1)\\times n}}\n     \\rho_1(A_{m})\n     \\cdot  \\left(\\int_{\\|a_m^\\perp\\|\\leq 1\/\\xi}\n     \\rho_2(a_m)\\, da_m\\right) dA_{m}.\\nonumber\n\\end{eqnarray}\nTo complete the proof it is sufficient to show the bound\n\\begin{equation}\\label{eq:C4}\n    \\int_{\\|a_m^\\perp\\|\\leq\\frac1\\xi}\n    \\rho_2(a_m)\\, da_m\n    \\leq   \\frac{1}{(\\sqrt{2\\pi})^{n-m+1}} \\,\n               \\frac{{\\cal O}_{n-m}}{n-m+1}\\,\n               \\Big(\\frac{1}{\\sigma \\xi}\\Big)^{n-m+1}\n\\end{equation}\nfor fixed, linearly independent $a_1,\\ldots,a_{m-1}$ and $\\xi>0$.\n\nTo show~\\eqref{eq:C4} note that\n$a_m^\\perp\\sim N(\\bar{a}_m^\\perp,\\sigma^2\\mathrm{I})$\nin $S^\\perp\\simeq\\mathbb{R}^{n-m+1}$\nwhere $\\bar{a}_m^\\perp$\nis the orthogonal projection of $\\bar{a}_m$ onto\n$S^\\perp$.\nLet $B_r$ denote the ball of radius~$r$ in $\\mathbb{R}^p$ centered\nat the origin. It is easy to see that $\\mathsf{vol\\,} B_r = {\\cal O}_{p-1}r^p\/p$.\nFor any\n$\\bar{x}\\in\\mathbb{R}^p$ and any $\\sigma>0$ we have\n\\begin{eqnarray*}\n  \\mathop{\\rm Prob}_{x\\sim N(\\bar{x},\\sigma^2\\mathrm{I})}\\big\\{\\|x\\|\\leq \\varepsilon\\big\\}\n  &\\leq & \\mathop{\\rm Prob}_{x\\sim N(0,\\sigma^2\\mathrm{I})}\\big\\{\\|x\\|\\leq \\varepsilon\\big\\}\n  \\;=\\; \\frac{1}{(\\sigma\\sqrt{2\\pi})^p}\n          \\int_{\\|x\\|\\leq \\varepsilon} e^{-\\frac{\\|x\\|^2}{2\\sigma^2}} dx\\\\\n  &\\stackrel{\\scriptstyle x=\\sigma z}{=}&\n    \\frac{1}{(\\sqrt{2\\pi})^p}\n          \\int_{\\|z\\|\\leq \\frac{\\varepsilon}{\\sigma}}\n         e^{-\\frac{\\|z\\|^2}{2}} dz\\\\\n  &\\leq & \\frac{1}{(\\sqrt{2\\pi})^p}\\,\n               \\mathsf{vol\\,} B_{\\frac{\\varepsilon}{\\sigma}}\n  \\;=\\; \\frac{1}{(\\sqrt{2\\pi})^p}\n               \\Big(\\frac{\\varepsilon}{\\sigma}\\Big)^p\\,\\mathsf{vol\\,} B_1\\\\\n   &=& \\frac{1}{(\\sqrt{2\\pi})^p}\n               \\Big(\\frac{\\varepsilon}{\\sigma}\\Big)^p\\,\n               \\frac{{\\cal O}_{p-1}}{p}.\n\\end{eqnarray*}\nTaking $\\bar{x}=\\bar{a}_m^\\perp$,\n$\\varepsilon=\\frac{1}{\\xi}$, and $p=n-m+1$\nthe claim~\\eqref{eq:C4} follows.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\proofof{Proposition~\\ref{thm:main_tail}}\nThe proof is based on an idea in~\\cite{sst:06}.\nFor $A\\in\\mathbb{R}^{m\\times n}$ there exists $u_A\\in\\mathbb{S}^{m-1}$ such that\n$\\|A^{\\dagger}\\|=\\|A^{\\dagger}u_A\\|$.\nMoreover, for almost all $A$, the vector $u_A$ is uniquely determined\nup to sign.\nUsing the singular value decomposition it is easy to show that,\nfor all $v\\in\\mathbb{S}^{m-1}$,\n\\begin{equation}\\label{eq:sankar}\n  \\|A^{\\dagger}v\\| \\geq \\|A^{\\dagger}\\| \\cdot |u_A^{\\rm T} v|.\n\\end{equation}\n\nNow take $A~\\sim N(\\overline{A},\\sigma^2\\mathrm{I})$ and\n$v\\sim U(\\mathbb{S}^{m-1})$ independently.\nThen, for any $s\\in(0,1)$ and $t>0$ we have\n\\begin{align*}\n  \\mathop{\\rm Prob}_{A,v}\\big\\{\\|A^{\\dagger}v\\|\\geq\\, t & \\sqrt{1-s^2}\\big\\}\n  \\;\\geq\\;\n  \\mathop{\\rm Prob}_{A,v}\\Big\\{\\|A^{\\dagger}\\|\\geq t\\ \\&\\\n    |u_A^{\\rm T} v|\\geq \\sqrt{1-s^2}\\Big\\}\\\\\n  =\\; & \\mathop{\\rm Prob}_{A}\\big\\{\\|A^{\\dagger}\\|\\geq t\\big\\} \\cdot\n     \\mathop{\\rm Prob}_{A,v}\\Big\\{|u_A^{\\rm T} v|\\geq \\sqrt{1-s^2}\n    \\;\\Big|\\; \\|A^{\\dagger}\\| \\geq t\\Big\\}\\\\\n  \\geq\\; & \\mathop{\\rm Prob}_{A}\\big\\{\\|A^{\\dagger}\\|\\geq t\\big\\} \\cdot\n   \\sqrt{\\frac{2}{\\pi m}}\\,s^{m-1},\n\\end{align*}\nthe last line by Lemma~\\ref{lem:s} with $\\xi=\\sqrt{1-s^2}$.\nNow we use Proposition~\\ref{prop:bound1} with\n$\\xi=t\\sqrt{1-s^2}$ to deduce that\n\\begin{eqnarray}\\label{eq:bound2}\n  \\mathop{\\rm Prob}_{A}\\big\\{\\|A^{\\dagger}\\|\\geq t \\big\\}\n  &\\leq& \\sqrt{\\frac{\\pi m}{2}}\\;\n   \\frac{1}{s^{m-1}}\n  \\mathop{\\rm Prob}_{A,v}\\{\\|A^{\\dagger}v\\|\\geq t\\sqrt{1-s^2}\\}\\\\\n  &\\leq&  \\frac{\\sqrt{m}}{2s^{m-1}}\\;\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\frac{{\\cal O}_{n-m}}{n-m+1}\\,\n               \\Big(\\frac{1}{\\sigma t\\sqrt{1-s^2}}\\Big)^{n-m+1}.\n               \\nonumber\n\\end{eqnarray}\n\nWe next choose $s\\in(0,1)$ to minimize the bound above.\nTo do so amounts to maximize\n$(1-x)^{\\frac{n-m+1}{2}} x^{\\frac{m-1}{2}}$ where\n$x=s^2\\in(0,1)$, or yet, to maximize\n$$\n   g(x)=\\Big((1-x)^{\\frac{n-m+1}{2}}\n    x^{\\frac{m-1}{2}}\\Big)^{\\frac2n}\n     =(1-x)^{\\frac{n-m+1}{n}} x^{\\frac{m-1}{n}}\n     =(1-x)^{1-\\lambda} x^\\lambda .\n$$\nWe have\n$\n   \\frac{d}{dx}\\ln g(x)= \\frac{\\lambda}{x}-\\frac{1-\\lambda}{1-x}\n$\nwith the only zero attained at $x^*=\\lambda$.\n\nReplacing $s^2$ by $\\lambda$ in~\\eqref{eq:bound2} we obtain the bound\n$$\n  \\mathop{\\rm Prob}_{A}\\big\\{\\|A^{\\dagger}\\|\\geq t \\big\\} \\ \\leq\\\n  \\frac{\\sqrt{\\lambda n+1}}\n  {2\\lambda^{\\frac{\\lambda n}{2}}}\\;\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\frac{{\\cal O}_{n-m}}{(1-\\lambda)n}\\,\n               \\bigg(\\frac{1}{\\sigma t\\sqrt{1-\\lambda}}\\bigg)^{(1-\\lambda)n} .\n$$\nLemma~\\ref{le:gbound} implies\n$$\n\\lambda^{-\\frac{\\lambda n}{2}}\n = \\Big( \\lambda^{-\\frac{\\lambda}{2(1-\\lambda)}}\\Big)^{(1-\\lambda)n}\n \\le e^{\\frac{(1-\\lambda)n}{2}} .\n$$\nSo we get\n\\begin{align*}\n\\mathop{\\rm Prob}_{A}\\big\\{& \\|A^{\\dagger}\\|\\geq t\\big\\} \\ \\leq\\\n\\frac{\\sqrt{\\lambda n+1}}{2}\\;\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\frac{{\\cal O}_{n-m}}{(1-\\lambda)n}\\,\n               \\bigg(\\frac{\\sqrt{e}}{\\sigma t\\sqrt{1-\\lambda}}\\bigg)^{(1-\\lambda)n} \\\\\n  \\;=\\;&\n  \\; \\frac{\\sqrt{\\lambda n +1}}{2}\\;\n    \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n       \\frac{1}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\frac{{\\cal O}_{n-m}}{(1-\\lambda)n}\\,\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\\\\\n =\\;& \\frac1{2(1-\\lambda)}\\;\n \\sqrt{\\lambda +\\frac1{n}}\\;\\frac1{\\sqrt{n}}\n\\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n       \\frac{{\\cal O}_{n-m}}{(\\sqrt{2\\pi})^{n-m}}\\,\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\\\\\n\\le\\;& \\frac{\\sqrt{\\lambda + 1}}{2(1-\\lambda)}\\;\\frac1{\\sqrt{n}}\n  \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n       \\frac{2\\pi^{\\frac{n-m+1}{2}}}\n       {\\Gamma\\big(\\frac{n-m+1}{2}\\big)(\\sqrt{2\\pi})^{n-m}}\\,\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\\\\\n = \\;& \\frac{\\sqrt{1+\\lambda}}{1-\\lambda}\\; \\frac1{\\sqrt{n}}\\;\n  \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n       \\frac{\\sqrt{2\\pi}}\n     {\\Gamma\\big(\\frac{n(1-\\lambda)}{2}\\big)2^{\\frac{(1-\\lambda)n}{2}}}\\,\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}.\n\\end{align*}\nWe next estimate $\\Gamma\\big(\\frac{(1-\\lambda)n}{2}\\big)$. To do so,\nrecall Stirling's bound\n$$\n  \\sqrt{2\\pi}x^{x+\\frac12}e^{-x}<\\Gamma(x+1)\n  <\\sqrt{2\\pi}x^{x+\\frac12}e^{-x+\\frac{1}{12x}}\n \\qquad\\mbox{for all $x>0$}\n$$\nwhich yields, using $\\Gamma(x+1)=x\\Gamma(x)$,\nthe bound $\\Gamma(x) > \\sqrt{2\\pi\/x}\\, (x\/e)^{x}$.\nWe use this with $x=\\frac{(1-\\lambda)n}{2}$ to obtain\n$$\n  \\Gamma\\Big(\\frac{(1-\\lambda)n}{2}\\Big)\n  \\geq \\sqrt{\\frac{4\\pi}{(1-\\lambda)n}}\\\n  \\Big(\\frac{(1-\\lambda)n}{2e}\\Big)^\\frac{(1-\\lambda)n}{2}.\n$$\nPlugging this into the above we obtain\n(observe the crucial cancellation of $\\sqrt{n}$)\n\\begin{align*}\n  \\mathop{\\rm Prob}_{A}\\big\\{&\\|A^{\\dagger}\\|\\geq t\\big\\} \\\\\n  \\leq\\ & \\sqrt{\\frac{1+\\lambda}{(1-\\lambda)^2}}\\; \\frac1{\\sqrt{n}}\\;\n  \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{\\frac{(1-\\lambda)n}{2}}\\;\n     \\, \\sqrt{2\\pi}\\, \\sqrt{\\frac{(1-\\lambda)n}{4\\pi}}\\,\n   \\Big(\\frac{e}{(1-\\lambda)n}\\Big)^\\frac{(1-\\lambda)n}{2}\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\\\\\n  =\\ &\n  c(\\lambda)\\, \\bigg(\\frac{e}{1-\\lambda}\\bigg)^{(1-\\lambda)n}\\;\n   \\Big(\\frac{1}{n}\\Big)^{\\frac{(1-\\lambda)n}{2}}\n               \\bigg(\\frac{1}{\\sigma t}\\bigg)^{(1-\\lambda)n}\n  = c(\\lambda)\\,\\bigg(\\frac{e}{\\sigma\\sqrt{n}(1-\\lambda)t}\\bigg)^{(1-\\lambda)n}\\;              ,\n\\end{align*}\nwhich completes the proof of the proposition.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\subsection{Proof of Theorem~\\ref{thm:tailbound}}\n\nTo simplify notation we write\n$c:=c(\\lambda)$ and $Q:=Q(m,n)$.\nProposition~\\ref{thm:main_tail} implies that for any $\\varepsilon>0$ we have\n\\begin{equation}\\label{eq:e_Adagger}\n\\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\Big\\{\n  \\|A^\\dagger\\|\\geq \\frac{e}{1-\\lambda}\\,\n    \\frac{1}{\\sigma\\sqrt{n}}\\,\n   \\Big(\\frac{c}{\\varepsilon}\\Big)^{\\frac{1}{(1-\\lambda)n}} \\Big\\}\n   \\ \\leq\\ \\varepsilon.\n\\end{equation}\nSimilarly, letting $\\varepsilon=e^{-\\frac{t^2}{2\\sigma^2}}$ in\nProposition~\\ref{le:enorm}\nand solving for $t$ we deduce that, for any $\\varepsilon\\in(0,1]$,\n\\begin{equation}\\label{eq:e_A}\n\\mathop{\\rm Prob}\\Big\\{\\|A\\|\\geq  Q\\sigma\\sqrt{n}+\\sigma\\sqrt{2\\ln\\frac{1}{\\varepsilon}} + 1\n     \\Big\\}\\ \\leq\\ \\varepsilon.\n\\end{equation}\nWe conclude that\n\\begin{equation}\\label{eq:tail_kappa}\n   \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\Big\\{\\kappa(A)\\geq\n   \\frac{ez(\\varepsilon)}{1-\\lambda}\\Big\\}\n   \\ \\leq\\ 2\\varepsilon,\n\\end{equation}\nwhere we have have set, for $\\varepsilon\\in(0,1]$,\n\\begin{equation}\\label{def:z(e)}\n  z(\\varepsilon):=\n     \\Bigg(Q + \\sqrt{\\frac{2}{n} \\ln\\frac1{\\varepsilon}} + \\frac1{\\sigma\\sqrt{n}}\\Bigg)\\,\n        \\Big(\\frac{c}{\\varepsilon}\\Big)^{\\frac{1}{(1-\\lambda)n}}.\n\\end{equation}\nWe note that $z(1)=\\zeta:=\\zeta_\\sigma(m,n)$, cf.~Equation~\\eqref{eq:defB}.\nMoreover, $\\lim_{\\varepsilon\\to 0}z(\\varepsilon) =\\infty$ and $z$ is decreasing in the\ninterval $(0,1]$.\nHence, for $z\\ge \\zeta$, there exists $\\varepsilon=\\varepsilon(z)\\in (0,1]$ such that $z=z(\\varepsilon)$.\n\nWe need to upper bound $\\varepsilon(z)$ as a function of $z$.\nTo do so, we start with a weak lower bound on $\\varepsilon(z)$ and\nclaim that\n\\begin{equation}\\label{eq:claim}\n\\frac1{n}\\,\\ln\\frac1{\\varepsilon}\\ \\le\\ \\ln (2z(\\varepsilon)) .\n\\end{equation}\nTo show this, recall that\n$Q\\ge \\sqrt{\\frac{n}{n+1}} \\ge \\frac1{\\sqrt{2}}$\ndue to Lemma~\\ref{pro:Espnorm}.\nHence\n$\\zeta \\ge Q \\ge 1\/\\sqrt{2}$ and it follows that\n$\\sqrt{2} z \\le 1$ for  $z \\ge \\zeta$.\nThus, Equation~\\eqref{def:z(e)} implies that\n$$\n z(\\varepsilon)\\ \\ge\\ \\frac{1}{\\sqrt{2}}\\,\n  \\Big(\\frac{c}{\\varepsilon}\\Big)^{\\frac{1}{(1-\\lambda)n}}.\n$$\nUsing $c\\ge \\frac1{\\sqrt{2}}$ we get\n$$\n (\\sqrt{2} z)^n \\ \\ge\\ (\\sqrt{2} z)^{(1-\\lambda)n}\n \\ge\\ \\frac{c}{\\varepsilon}\\ \\ge\\ \\frac1{\\sqrt{2}\\,\\varepsilon}.\n$$\nHence $(2z)^n \\ge 1\/\\varepsilon$, which shows\nthe claimed inequality~\\eqref{eq:claim}.\n\nUsing the bound~\\eqref{eq:claim} in Equation~\\eqref{def:z(e)} we get,\nagain writing $z=z(\\varepsilon)$, that\n$$\n z\\ \\le\\\n \\Big(Q + \\sqrt{2\\ln (2z)} + \\frac1{\\sigma\\sqrt{n}}\\Big)\\,\n        \\Big(\\frac{c}{\\varepsilon}\\Big)^{\\frac{1}{(1-\\lambda)n}},\n$$\nwhich means\n$$\n \\varepsilon\\ \\le\\\n  c \\bigg[\n   \\Big(Q + \\sqrt{2\\ln(2z)} + \\frac1{\\sigma\\sqrt{n}}\\Big)\\,\n    \\frac1z\\bigg]^{(1-\\lambda)n}.\n$$\nBy~\\eqref{eq:tail_kappa} this completes the proof.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\subsection{Proof of Corollary~\\ref{cor:main}}\n\nFix $\\lambda_0\\in (0,1)$ and put $c:=c(\\lambda_0)$.\nSuppose that $m\\le n$ satisfy $\\lambda =(m-1)\/n \\le \\lambda_0$.\nThen $n-m +1 = (1-\\lambda)n \\ge (1-\\lambda_0) n$ and\nin order to have $n-m$ sufficiently large\nit suffices to require that $n$ is sufficiently large.\nThus, $c^{\\frac1{n-m+1}} \\le 1.1$\nif $n$ is sufficiently large. Similarly, because of\nLemma~\\ref{pro:Espnorm},\n$Q(m,n)\\leq 2.1$\nfor large enough~$n$. This implies that,\nfor $\\frac1{\\sqrt{m}} \\le \\sigma\\le 1$,\nwe have\n$$\n Q(m,n)  + \\frac1{\\sigma\\sqrt{n}}\\ \\le\\\n 2.1 + \\frac1{\\sigma\\sqrt{n}} \\ \\le\\\n 2.1 + \\sqrt{\\frac{m}{n}} \\ \\le\\\n 2.1 + \\sqrt{\\lambda_0 +\\frac1{n}}\\ \\le\\ 3.1 ,\n$$\nprovided $n$ is large enough. Then\n$\\zeta_\\sigma(m,n) \\le 3.1\\cdot 1.1 = 3.41$.\n\nBy Theorem~\\ref{thm:tailbound},\nthe random variable\n$Z :=  (1-\\lambda)\\kappa(A)\/e$\nsatisfies, for any $\\overline{A}$ with $\\|\\overline{A}\\|\\le 1$ and\nany $z\\ge 3.41$,\n\\begin{eqnarray*}\n \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\big\\{ Z \\ge z \\big\\}\n  &\\leq& 2c\\,\\bigg[\\Big(Q(m,n)+ \\sqrt{2\\ln (2z)} + \\frac1{\\sigma\\sqrt{n}}\\Big)\\,\n    \\frac1z\\bigg]^{n-m+1} \\\\\n  &\\leq& 2c\\,\\bigg[\\Big(3.1+ \\sqrt{2\\ln (2z)}\\Big)\\,\n    \\frac1z\\bigg]^{n-m+1}.\n\\end{eqnarray*}\nSince\n$3.1 + \\sqrt{2\\ln (2z)}\\le e\\sqrt{z}$\nfor $z\\ge 4$\nwe deduce that, for all such $z$,\n$$\n \\mathop{\\rm Prob}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}\\big\\{ Z \\ge z \\big\\}\n  \\ \\leq\\ 2c\\,\\Big(\\frac{e}{\\sqrt{z}}\\Big)^{n-m+1} .\n$$\nUsing this tail bound to compute $\\mathbb{E}(Z)$ we get\n\\begin{eqnarray*}\n \\mathbb{E}(Z) &=& \\int_0^\\infty \\mathop{\\rm Prob}\\{Z\\ge z\\}\\, dz\\\n  \\;\\le\\; e^2 + 2c\n  \\int_{e^2}^\\infty\n  \\left(\\frac{e^2}{z}\\right)^{\\frac{n-m+1}{2}}dz \\\\\n  &\\stackrel{\\scriptstyle z=e^2y}{=}&\n    e^2 + 2c\\int_1^\\infty\n  \\left(\\frac{1}{y}\\right)^{\\frac{n-m+1}{2}}e^2dy\n  \\;=\\; e^2+ \\frac{4ce^2}{n-m-1}.\n\\end{eqnarray*}\nWe can now conclude since\n$$\n \\mathbb{E}((1-\\lambda)\\kappa(A))\n  = \\mathbb{E}(eZ) =e\\mathbb{E}(Z)\\leq e^3+ \\frac{4ce^3}{n-m-1}\n  \\leq 20.1\n$$\nthe inequality, again, by taking $n$ large enough.\n{\\mbox{}\\hfill\\qed}\\medskip\n\n\\section{Applications}\n\\label{sec:applications}\n\nWe next briefly discuss the two applications of our main result\nmentioned in the introduction.\n\n\\subsection{Accuracy of Linear Least Squares}\n\nRecall the problem~\\eqref{eq:ls} described in the introduction, namely,\nto compute the minimum of $\\|Ax-b\\|^2$ over $x\\in\\mathbb{R}^n$ \nfor given $A\\in\\mathbb{R}^{m\\times n}$ and $b\\in\\mathbb{R}^m$ (with $m>n$).\nIt is well known\nthat the loss of precision $\\mathsf{LoP}(A^\\dagger b)$\n---that is, the number of correct digits in\nthe entries of the data $(A,b)$ minus the same number for the computed\nsolution $A^\\dagger b$--- satisfies\n(cf.~\\cite{Wedin73} and \\cite[Ch.~19]{higham:96}) \n$$\n  \\mathsf{LoP}(A^\\dagger b)\\leq \\log m n^{3\/2} + 2\\log\\kappa(A)+{\\cal O}(1).\n$$\n Corollary~\\ref{cor:main}, combined with Jensen's inequality,\nimplies that $\\mathbb{E}(\\log\\kappa(A)) \\le \\log(20.1\/(1-\\lambda)) = {\\cal O}(1)$\nunder the assumptions stated in the corollary.\nHence for sufficiently elongated, large matrices $\\overline{A}$, the\nexpected loss of precision in the computation of the solution\n$A^\\dagger b$ over all small perturbations $A$ of $\\overline{A}$ is\ndominated by the term $\\log m n^{3\/2}$.\n\n\\subsection{Complexity of the Conjugate Gradient Method}\n\nIf $P\\in\\mathbb{R}^{m\\times m}$ is a symmetric positive definite\nmatrix and $c\\in\\mathbb{R}^m$, the system $Px=c$ can be solved by\nthe Conjugate Gradient Method (CGM), cf.~\\cite{hest-stiefel:52}.\nThis is an iterative algorithm which performs at most $m$ iterations but\nmay require less. Indeed, it is known\n(see, e.g., \\cite[Lecture~38]{ThrefethenBau})\nthat an $\\varepsilon$-approximation\nof the solution~$x$ can be computed in at most\n$\\frac12\\sqrt{\\kappa(P)}|\\ln\\varepsilon| $\niterations\n($\\varepsilon$ measures the\nrelative error of the approximation with respect to the\nEuclidean norm).\n\nIn many cases the matrix $P$ arises as $P=AA^{\\rm T}$ for some\nmatrix $A\\in\\mathbb{R}^{m\\times n}$ with $n>m$. \nIf $A$ is standard Gaussian distributed, \nthen the resulting distribution of $P$, \ncalled {\\em Wishart distribution}, \nhas been extensively studied in multivariate statistics. \nHowever, in our case of interest, $A$ is noncentered \nand much less is known about the resulting \ndistribution of $P$ \n(called noncentral Wishart). \nFortunately, using the fact that $\\sqrt{\\kappa(P)}=\\kappa(A)$, \nwe can directly apply our tail bounds for $\\kappa(A)$ \nfor a noncentral, isotropic Gaussian \ndistribution of $A$,  \nto derive bounds for the expected number\nof iterations of CGM.\n\nTo do so we use again Corollary~\\ref{cor:main}.\nIt shows that for all $\\lambda_0\\in(0, 1)$ and all\n$0 <\\sigma \\le 1$ there exists $n_0$ such that for\nall $1\\leq m<n$ we have\n$$\n \\sup_{\\|\\overline{A}\\| \\le 1} \\mathbb{E}_{A\\sim N(\\overline{A},\\sigma^2\\mathrm{I})}(\\kappa(A))\n  \\ \\leq\\ \\frac{20.1}{1-\\lambda},\n$$\nprovided $\\lambda = \\frac{m-1}{n}\\leq \\lambda_0$ and $n\\geq n_0$.\nIt follows that if $P$ is obtained as $AA^{\\rm T}$ for a large, elongated,\nrectangular matrix $A$ then, we should expect to\ncompute a solution with the desired accuracy with\nabout $\\frac{1}{2}\\frac{20.1}{1-\\lambda}|\\ln\\varepsilon|$ iterations.\nIt is known that\neach iteration of CGM takes $6n^2 +{\\cal O}(n)$ arithmetic operations.\nTherefore, the expected cost of running CGM on $P$ is\n$$\n  3n^2 \\frac{20.1}{1-\\lambda}|\\ln\\varepsilon|\n +{\\cal O}(n)=\\frac{60.3n^2}{1-\\lambda}|\\ln\\varepsilon| +{\\cal O}(n).\n$$\nThe leading term in this expression is smaller than the\n$\\frac23 n^3$ operations performed\nby Gaussian elimination as long as\n$$\n  \\varepsilon\\geq e^{-\\frac{n(1-\\lambda)}{91}}.\n$$\nFor large $n$ (and $\\lambda$ not too close to 1) this bound produces\nvery small values of $\\varepsilon$\nand therefore, CGM yields, on the average (both for a Wishart distribution\nof data $P$ and for Wishart perturbations of arbitrary data), \nremarkably good approximations of the solution $x=P^{-1}c$.\n\n\\section{Some Numerical Simulations}\n\\label{sec:numerical}\n\nSection~6 in~\\cite{CDW:05} describes the result of numerical computations\nproducing experimental values for $\\mathbb{E}(\\ln\\kappa(A))$ (for certain\nchoices of $\\overline{A}$ and $\\sigma$), which are\ndenoted by $\\mathbf{Avr}(\\ln\\kappa(A))$ and compared with\nthe upper bound for $\\mathbb{E}(\\ln\\kappa(A))$\n$$\n  {\\mu(m,n,\\sigma)}:=\n  \\ln\\left(m\n  +\\sigma m\\sqrt{5n}\\right)\n  +\\ln \\frac{2.35}{\\sigma} + \\frac{1}{r}\n  +\\sqrt{\\frac{e\\pi}{5}}\n$$\nobtained there. The data in\nTables~\\ref{tab:2} to~\\ref{tab:5}\nis taken from~\\cite{CDW:05}.\nEach row in these tables corresponds to a pair $(m,n)$.\nFor each row, 500 random matrices $A\\in\\mathbb{R}^{m\\times n}$\nwere computed following the distribution $N(\\sqrt{m}\\,\\overline{A},\\mathrm{I})$,\nwhere $\\overline{A}$ was chosen as\n$$\n  \\overline{A} := \\frac{{\\sf ones(m,n)}}{\\|{\\sf ones(m,n)}\\|},\n$$\nand {\\sf ones(m,n)} denotes the $m\\times n$ matrix all of whose\nentries are 1.\nThe column with header ${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ shows the empirical average of\n$\\ln\\kappa(A)$ for the 500 chosen random matrices~$A$.\nSince $\\kappa(A)$ is scale invariant we note that this corresponds\nto random matrices chosen from $N(\\overline{A},\\sigma\\mathrm{I})$,\nwhere $\\sigma=1\/\\sqrt{m}$.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{||c|c|c|c|c||}\n \\hline\n $m$ & $n$ &${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ & ${\\mu(m,n,\\sigma)}$ & $\\ln (20.1\/(1-\\lambda))$\\\\\n\\hline \\hline\n10&15& 1.88278226808667 &7.73190477060415 & \\\\\n\\cline{1-4} 20&30&  2.04718612539162 & 8.74083698937094 &\n\\\\\n\\cline{1-4}\n40&60 &2.13539482051851&9.75820027818245& 4.0993321\n\\\\\n\\cline{1-4}\n80 & 120 & 2.19377719811291 & 10.78180469776403 &\n\\\\\n \\cline{1-4}\n160& 240&2.23119383890675&11.80997066079053&\n\\\\\n\\hline\n\\end{tabular}\\caption{$n=1.5m.$}\\label{tab:2}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{||c|c|c|c|c||}\n \\hline\n $m$ & $n$ &${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ & ${\\mu(m,n,\\sigma)}$ & $\\ln (20.1\/(1-\\lambda))$\\\\\n\\hline \\hline\n 5& 10 & 1.28204418194521&6.35902343647518&\\\\\n\\cline{1-4}\n10&20& 1.48669849397793 &7.36178009761038&  \\\\\n\\cline{1-4}\n20&40 &  1.59394635398509&8.37451330180407 & \\\\\n \\cline{1-4}\n 40& 80&1.64896402420115& 9.39470162365532& 3.693866\\\\\n \\cline{1-4}\n80&160&1.69565973841311&10.42037692088400&\\\\\n \\cline{1-4}\n 160 &320 &1.72154032592663 &11.45004561375610&\\\\\n \\hline\n\\end{tabular} \\caption{$n=2m.$}\\label{tab:3}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{||c|c|c|c|c||}\n \\hline\n $m$ & $n$ &${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ & ${\\mu(m,n,\\sigma)}$ & $\\ln (20.1\/(1-\\lambda))$\\\\\n\\hline \\hline\n10&25& 1.24167342192086 &7.46370799208199&\\\\\n\\cline{1-4}\n20&50 &1.34213347902230&8.47908853717777 &  \\\\\n \\cline{1-4}\n 40& 100&1.40120155287858&9.50123344342563& 3.511545 \\\\\n \\cline{1-4}\n80&200&1.44120596017225&10.52833707967242&\\\\\n \\cline{1-4}\n 160 &400 &1.45928497502137&11.55903912197539&\\\\\n \\hline\n\\end{tabular} \\caption{$n=2.5 m.$}\\label{tab:4}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{||c|c|c|c|c||}\n \\hline\n $m$ & $n$ &${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ & ${\\mu(m,n,\\sigma)}$ & $\\ln (20.1\/(1-\\lambda))$\\\\  \\hline \\hline\n 5& 15 &  0.98741849882614& 6.37209092337754 &  \\\\\n\\cline{1-4}\n10&30&1.10550395287499 &7.38102314214432&  \\\\\n\\cline{1-4}\n20&60 &1.18790345922560&8.39838643095583 & \\\\\n \\cline{1-4}\n 40& 120& 1.23914387557043&9.42199085053742& 3.406185 \\\\\n \\cline{1-4}\n80&240& 1.27096561714092&10.45015681356392&\\\\\n \\cline{1-4}\n 160 &480  &1.28600775609989&12.14829242876138&\\\\\n \\hline\n\\end{tabular}\\caption{$n=3m.$}\\label{tab:5}\n\\end{center}\n\\end{table}\n\nIn~\\cite{CDW:05} it is\nobserved that ``one sees that when one fixes $m$ and lets $n$ increase\nthe quantity $\\mathbf{Avr}(\\ln\\kappa(A))$ decreases. This is in contrast\nwith the behaviour of $\\mu(m,n,\\sigma)$. It appears that our methods\nare not sharp enough to capture the behaviour of\n$\\mathbb{E}(\\ln\\kappa(A))$.'' Compare now with the\nresults of the present paper. It follows from\nCorollary~\\ref{cor:main}, by Jensen's inequality, that,\nfor sufficiently large $n$,\n$\\mathbb{E}(\\ln\\kappa(A))\\leq \\ln \\frac{20.1}{1-\\lambda}$. But if $m$ is\nheld fixed then, when $n$ increases,\n$\\lambda$ decreases and so does $\\ln \\frac{20.1}{1-\\lambda}$.\n\nOne still observes a difference between the bound $\\ln \\frac{20.1}{1-\\lambda}$\nand the values of ${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$. Part of this difference comes from the\nasymptotic character of this bound and the fact that our data\nis limited to $m\\leq 160$. One sees on the tables that larger values\nof $m$ would approach ${{\\mathbf{Avr}\\left(\\ln\\kappa(A)\\right)}}$ to $\\ln \\frac{20.1}{1-\\lambda}$.\nWe conjecture that, in addition to the possible loss of sharpness\ncoming from the use of Jensen's inequality, the difference above is due to\nthe roughness of the constant 20.1.\n\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}